Basics (and advanced stuff) of questionnaire development and analysis
Julius Fenn
26 04 2021
- 1 show data set
- 2 [simulation study: standardized residuals, reliability]
- 3 developing a questionnaire scale
- References
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1 show data set
Datensatz ist der Campus-File des IQB-Ländervergleichs 2011 der Primarstufe (Zugang über ), Bedeutung Variablen einsichtig über Suchfunktion Skalenhandbuch.
dim(datenLV)## [1] 3005 33knitr::kable(datenLV[1:4,], digits = 2)| idsch_FDZ | idstud_FDZ | tr_sex | tr_age | Emigr | EDezh | EHisei | EHisced_akt | SBuecher | SLesZt | tr_NotDe | tr_NotMa | tr_Wdh_r | SSkDe_a | SSkDe_b | SSkDe_c | SSkDe_d | SSkMa_a | SSkMa_b | SSkMa_c | SSkMa_d | SBezMs_a | SBezMs_b | SBezMs_c | SBezMs_d | SSkDe | SSkMa | SBezMs | wle_lesen | wle_hoeren | wle_mathe | schoolEconDis | schoolMiganteil |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | maennlich | 10.42 | keinMig | Kind spricht zu Hause immer oder fast immer Deutsch | 55 | ISCED level 5B | 100 Buecher | 30 Minuten bis zu einer Stunde | 2 | 2 | nein | 4 | 1 | 4 | 4 | 3 | 3 | 3 | 4 | 4 | 3 | 1 | 1 | 4.00 | 3.00 | 3.75 | -0.17 | -0.63 | -0.33 | 33-66% oekonomisch mittel | > 20% Miganteil |
| 1 | 2 | maennlich | 9.83 | Mig | Kind spricht zu Hause immer oder fast immer Deutsch | 49 | ISCED level 5B | mehr 200 Buecher | 30 Minuten bis zu einer Stunde | 2 | 2 | nein | 2 | 2 | 2 | 2 | 3 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 2.25 | 3.25 | 3.50 | -0.44 | -0.98 | 0.67 | 33-66% oekonomisch mittel | > 20% Miganteil |
| 1 | 3 | maennlich | 10.50 | NA | Kind spricht zu Hause nie oder manchmal Deutsch | NA | NA | 25 Buecher | 30 Minuten bis zu einer Stunde | 4 | 3 | nein | 3 | 1 | 4 | 3 | 3 | 1 | 3 | 3 | 4 | 1 | 1 | 2 | 3.50 | 3.25 | 3.00 | -1.21 | -1.06 | -0.67 | 33-66% oekonomisch mittel | > 20% Miganteil |
| 1 | 4 | maennlich | 10.67 | NA | NA | NA | NA | 10 Buecher | weniger als 30 Minuten | 2 | 3 | nein | 3 | 3 | 4 | 4 | 4 | 1 | 4 | 4 | 1 | 4 | 4 | 4 | 3.25 | 4.00 | 1.75 | 0.26 | -1.17 | 0.14 | 33-66% oekonomisch mittel | > 20% Miganteil |
summary(datenLV)## idsch_FDZ idstud_FDZ tr_sex tr_age
## Min. : 1.0 Min. : 1 maennlich:1530 Min. : 6.833
## 1st Qu.: 53.0 1st Qu.: 752 weiblich :1475 1st Qu.:10.083
## Median :104.0 Median :1503 Median :10.417
## Mean :103.3 Mean :1503 Mean :10.425
## 3rd Qu.:155.0 3rd Qu.:2254 3rd Qu.:10.750
## Max. :201.0 Max. :3005 Max. :13.000
## NA's :7
## Emigr EDezh
## Mig : 493 Kind spricht zu Hause immer oder fast immer Deutsch:2346
## keinMig:1966 Kind spricht zu Hause nie oder manchmal Deutsch : 191
## NA's : 546 NA's : 468
##
##
##
##
## EHisei EHisced_akt SBuecher
## Min. :10.00 ISCED level 1 : 23 10 Buecher : 153
## 1st Qu.:37.00 ISCED level 2 : 81 25 Buecher : 557
## Median :48.00 ISCED level 3A: 10 100 Buecher :1112
## Mean :49.57 ISCED level 5B:1556 200 Buecher : 563
## 3rd Qu.:61.00 ISCED level 5A: 724 mehr 200 Buecher: 550
## Max. :89.00 ISCED level 6 : 126 NA's : 70
## NA's :622 NA's : 485
## SLesZt tr_NotDe tr_NotMa
## weniger als 30 Minuten : 862 Min. :1.00 Min. :1.000
## 30 Minuten bis zu einer Stunde:1173 1st Qu.:2.00 1st Qu.:2.000
## 1-2 Stunden : 475 Median :2.00 Median :2.000
## 2 Stunden oder mehr : 398 Mean :2.46 Mean :2.509
## NA's : 97 3rd Qu.:3.00 3rd Qu.:3.000
## Max. :5.00 Max. :5.000
## NA's :131 NA's :127
## tr_Wdh_r SSkDe_a SSkDe_b SSkDe_c SSkDe_d
## nein:2830 Min. :1.000 Min. :1.000 Min. :1.000 Min. :1.000
## ja : 168 1st Qu.:3.000 1st Qu.:1.000 1st Qu.:3.000 1st Qu.:3.000
## NA's: 7 Median :3.000 Median :2.000 Median :3.000 Median :3.000
## Mean :3.116 Mean :2.126 Mean :3.303 Mean :3.334
## 3rd Qu.:4.000 3rd Qu.:3.000 3rd Qu.:4.000 3rd Qu.:4.000
## Max. :4.000 Max. :4.000 Max. :4.000 Max. :4.000
## NA's :92 NA's :121 NA's :126 NA's :123
## SSkMa_a SSkMa_b SSkMa_c SSkMa_d SBezMs_a
## Min. :1.00 Min. :1.000 Min. :1.000 Min. :1.000 Min. :1.000
## 1st Qu.:3.00 1st Qu.:1.000 1st Qu.:3.000 1st Qu.:3.000 1st Qu.:3.000
## Median :3.00 Median :2.000 Median :3.000 Median :4.000 Median :3.000
## Mean :3.17 Mean :2.065 Mean :3.314 Mean :3.331 Mean :3.365
## 3rd Qu.:4.00 3rd Qu.:3.000 3rd Qu.:4.000 3rd Qu.:4.000 3rd Qu.:4.000
## Max. :4.00 Max. :4.000 Max. :4.000 Max. :4.000 Max. :4.000
## NA's :82 NA's :115 NA's :121 NA's :102 NA's :164
## SBezMs_b SBezMs_c SBezMs_d SSkDe SSkMa
## Min. :1.00 Min. :1.000 Min. :1.000 Min. :1.000 Min. :1.000
## 1st Qu.:3.00 1st Qu.:1.000 1st Qu.:1.000 1st Qu.:2.750 1st Qu.:2.750
## Median :3.00 Median :1.000 Median :1.000 Median :3.250 Median :3.250
## Mean :3.11 Mean :1.633 Mean :1.493 Mean :3.156 Mean :3.187
## 3rd Qu.:4.00 3rd Qu.:2.000 3rd Qu.:2.000 3rd Qu.:3.750 3rd Qu.:4.000
## Max. :4.00 Max. :4.000 Max. :4.000 Max. :4.000 Max. :4.000
## NA's :199 NA's :173 NA's :172 NA's :95 NA's :85
## SBezMs wle_lesen wle_hoeren wle_mathe
## Min. :1.000 Min. :-5.06686 Min. :-5.8078 Min. :-3.4768
## 1st Qu.:3.000 1st Qu.:-0.68091 1st Qu.:-0.5796 1st Qu.:-0.6384
## Median :3.500 Median : 0.12715 Median : 0.1526 Median : 0.1035
## Mean :3.335 Mean : 0.09367 Mean : 0.1094 Mean : 0.1061
## 3rd Qu.:3.750 3rd Qu.: 0.88132 3rd Qu.: 0.8188 3rd Qu.: 0.8379
## Max. :4.000 Max. : 4.24000 Max. : 3.5043 Max. : 4.7832
## NA's :145
## schoolEconDis schoolMiganteil
## <33% oekonomisch benachteiligt: 175 < 20% Miganteil:1645
## 33-66% oekonomisch mittel :2320 > 20% Miganteil:1360
## >66% oekonomisch bevorzugt : 510
##
##
##
## 1.1 tidy and transform data
psych::corPlot(r = datenLV[,str_subset(string = colnames(datenLV), pattern = "^SSkMa_|^SSkDe_|^SBezMs_")])
# recode
datenLV$SSkMa_b <- 5 - datenLV$SSkMa_b
datenLV$SSkDe_b <- 5 - datenLV$SSkDe_b
datenLV$SBezMs_c <- 5 - datenLV$SBezMs_c
datenLV$SBezMs_d <- 5 - datenLV$SBezMs_d
psych::corPlot(r = datenLV[,str_subset(string = colnames(datenLV), pattern = "^SSkMa_|^SSkDe_|^SBezMs_")])
1.2 describe, visualize your data and get a first impression using R base functions
Insgesamt weisst der Datensatz N=3005 Schüler/innen in 201 Schulen auf. Die Schulen weisen folgende Anzahl von Schüler/innen auf:
proSchule <- aggregate(datenLV$idsch_FDZ,by=list(datenLV$idsch_FDZ),FUN=length) # using base functions
summary(proSchule$x)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 5.00 10.00 15.00 14.95 20.00 20.00rm(proSchule)plot(datenLV$wle_lesen, datenLV$wle_mathe)
boxplot(datenLV$wle_lesen ~ datenLV$Emigr)
t.test(datenLV$wle_lesen ~ datenLV$Emigr) # = unequal variances t-test##
## Welch Two Sample t-test
##
## data: datenLV$wle_lesen by datenLV$Emigr
## t = -8.6373, df = 720.46, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -0.6554145 -0.4126442
## sample estimates:
## mean in group Mig mean in group keinMig
## -0.2612065 0.2728229Bestehen lineare Zusammenhänge mit einer (normalverteilten) numerischen Variablen?
## linear regression
summary(lm(formula = wle_lesen ~ Emigr*tr_sex + SSkMa, data = datenLV))##
## Call:
## lm(formula = wle_lesen ~ Emigr * tr_sex + SSkMa, data = datenLV)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.3692 -0.7007 0.0184 0.7098 3.8719
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.98582 0.12494 -15.894 < 2e-16 ***
## EmigrkeinMig 0.39521 0.07802 5.065 4.38e-07 ***
## tr_sexweiblich 0.30605 0.10250 2.986 0.00286 **
## SSkMa 0.52233 0.03229 16.174 < 2e-16 ***
## EmigrkeinMig:tr_sexweiblich 0.04767 0.11380 0.419 0.67535
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.108 on 2399 degrees of freedom
## (601 observations deleted due to missingness)
## Multiple R-squared: 0.1324, Adjusted R-squared: 0.1309
## F-statistic: 91.49 on 4 and 2399 DF, p-value: < 2.2e-16Bestehen lineare Zusammenhänge mit einer binären Variablen? empfohlene Seite: https://www.methodenberatung.uzh.ch/de/datenanalyse_spss/zusammenhaenge/lreg.html
## logistic regression (if > 2 -> ordinal logistic regression)
summary(glm(Emigr ~ wle_lesen+wle_hoeren+wle_mathe,
data = datenLV, family = binomial))##
## Call:
## glm(formula = Emigr ~ wle_lesen + wle_hoeren + wle_mathe, family = binomial,
## data = datenLV)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.3848 0.4196 0.5748 0.6953 1.4128
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 1.36913 0.05226 26.198 < 2e-16 ***
## wle_lesen 0.17045 0.05383 3.167 0.00154 **
## wle_hoeren 0.12008 0.05871 2.045 0.04081 *
## wle_mathe 0.33122 0.05930 5.585 2.33e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 2464.3 on 2458 degrees of freedom
## Residual deviance: 2338.4 on 2455 degrees of freedom
## (546 observations deleted due to missingness)
## AIC: 2346.4
##
## Number of Fisher Scoring iterations: 4exp(coef(glm(Emigr ~ wle_lesen+wle_hoeren+wle_mathe,
data = datenLV, family = binomial))) - 1## (Intercept) wle_lesen wle_hoeren wle_mathe
## 2.9319093 0.1858348 0.1275879 0.3926618Anmerkung: Hypothestentest, logistische Regression sind die zentralen Verfahren für die deduktive Methode der Itementwicklung
1.3 missing data
## missing data patterns
mdpattern <- mice::md.pattern(x = datenLV[,str_subset(string = colnames(datenLV), pattern = "^SSkMa_|^SSkDe_")], plot = TRUE, rotate.names = TRUE)
## sorted by # missing
mdpattern[match(x = sort(x = as.numeric(rownames(mdpattern)), decreasing = TRUE), table = as.numeric(rownames(mdpattern))), ]## SSkMa_a SSkDe_a SSkMa_d SSkMa_b SSkDe_b SSkMa_c SSkDe_d SSkDe_c
## 2756 1 1 1 1 1 1 1 1 0
## 50 0 0 0 0 0 0 0 0 8
## 22 1 0 1 1 0 1 0 0 4
## 16 1 1 1 1 1 1 1 0 1
## 16 1 1 1 1 1 1 1 0 1
## 15 1 1 1 0 1 1 1 1 1
## 14 1 1 1 1 1 1 0 1 1
## 12 1 1 1 1 1 0 1 1 1
## 11 0 1 0 0 1 0 1 1 4
## 10 0 1 1 1 1 1 1 1 1
## 7 1 1 1 0 1 0 1 1 2
## 6 1 1 0 0 1 0 1 1 3
## 5 1 1 0 1 1 1 1 1 1
## 4 1 1 1 1 1 1 0 0 2
## 4 1 1 1 1 1 1 0 0 2
## 4 1 1 1 1 1 1 0 0 2
## 3 1 1 1 0 0 1 1 1 2
## 3 1 1 1 0 0 1 1 1 2
## 3 1 1 1 0 0 1 1 1 2
## 3 1 1 1 0 0 1 1 1 2
## 2 1 1 1 1 1 0 1 0 2
## 2 1 1 1 1 1 0 1 0 2
## 2 1 1 1 1 1 0 1 0 2
## 2 1 1 1 1 1 0 1 0 2
## 2 1 1 1 1 1 0 1 0 2
## 2 1 1 1 1 1 0 1 0 2
## 2 1 1 1 1 1 0 1 0 2
## 2 1 1 1 1 1 0 1 0 2
## 2 1 1 1 1 1 0 1 0 2
## 2 1 1 1 1 1 0 1 0 2
## 1 1 1 1 1 0 1 0 1 2
## 1 1 1 1 1 0 1 0 1 2
## 1 1 1 1 1 0 1 0 1 2
## 1 1 1 1 1 0 1 0 1 2
## 1 1 1 1 1 0 1 0 1 2
## 1 1 1 1 1 0 1 0 1 2
## 1 1 1 1 1 0 1 0 1 2
## 1 1 1 1 1 0 1 0 1 2
## 1 1 1 1 1 0 1 0 1 2
## 1 1 1 1 1 0 1 0 1 2
## 1 1 1 1 1 0 1 0 1 2
## 1 1 1 1 1 0 1 0 1 2
## 1 1 1 1 1 0 1 0 1 2
## 1 1 1 1 1 0 1 0 1 2
## 1 1 1 1 1 0 1 0 1 2
## 1 1 1 1 1 0 1 0 1 2
## 1 1 1 1 1 0 1 0 1 2
## 1 1 1 1 1 0 1 0 1 2
## 1 1 1 1 1 0 1 0 1 2
## 1 1 1 1 1 0 1 0 1 2
## 1 1 1 1 1 0 1 0 1 2## generate dummy of missing variable to identify potential helper variables
datenLV$missing_SSkMa_a <- ifelse(test = is.na(datenLV$SSkMa_a), yes = 1, no = 0)
helpervars <- c("wle_lesen", "wle_hoeren", "SSkDe") # include normally many more
for(v in helpervars){
tmp <- t.test(datenLV[[v]] ~ datenLV$missing_SSkMa_a)
if(tmp$p.value < .05){
print(v)
print(tmp)
}
}## [1] "wle_lesen"
##
## Welch Two Sample t-test
##
## data: datenLV[[v]] by datenLV$missing_SSkMa_a
## t = 4.0297, df = 84.449, p-value = 0.0001217
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 0.3177873 0.9369377
## sample estimates:
## mean in group 0 mean in group 1
## 0.1107931 -0.5165693
##
## [1] "wle_hoeren"
##
## Welch Two Sample t-test
##
## data: datenLV[[v]] by datenLV$missing_SSkMa_a
## t = 4.4992, df = 83.294, p-value = 2.193e-05
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 0.4094216 1.0581596
## sample estimates:
## mean in group 0 mean in group 1
## 0.1294574 -0.6043332## overall missing for each single variable
round(x = sort(x = colSums(x = is.na(datenLV), na.rm = TRUE), decreasing = TRUE) / nrow(datenLV) * 100, digits = 2)## EHisei Emigr EHisced_akt EDezh SBezMs_b
## 20.70 18.17 16.14 15.57 6.62
## SBezMs_c SBezMs_d SBezMs_a SBezMs tr_NotDe
## 5.76 5.72 5.46 4.83 4.36
## tr_NotMa SSkDe_c SSkDe_d SSkDe_b SSkMa_c
## 4.23 4.19 4.09 4.03 4.03
## SSkMa_b SSkMa_d SLesZt SSkDe SSkDe_a
## 3.83 3.39 3.23 3.16 3.06
## SSkMa SSkMa_a SBuecher tr_age tr_Wdh_r
## 2.83 2.73 2.33 0.23 0.23
## idsch_FDZ idstud_FDZ tr_sex wle_lesen wle_hoeren
## 0.00 0.00 0.00 0.00 0.00
## wle_mathe schoolEconDis schoolMiganteil missing_SSkMa_a
## 0.00 0.00 0.00 0.00Es ist zentral fehlende Daten zu ersetzen bzw. modellbasiert zu schätzen. Die zwei modernsten Ansätze, um fehlende Daten zu ersetzen sind:
- multiple imputation, introductory book: https://stefvanbuuren.name/fimd/, Grund, Lüdtke, and Robitzsch (2018)
- full information maximum likelihood (aktuell in Mplus, aber nicht in R implementiert)
1.4 outlier analysis
Es wird unterschieden in uni- und multivariate Ausreißer, da structural equation modelling / CFA multivariate Verfahren sind (mehrere UVs und AVs), ist es notwendig die Daten auf multivariate Ausreißer zu kontrollieren. Dafür eignet sich die Mahalanobis Distance:
## exemplify Mahalanobis Distance
sigma <- matrix(c(4,1,2,1,5,4,2,4,6), ncol = 3)
cov2cor(sigma)## [,1] [,2] [,3]
## [1,] 1.0000000 0.2236068 0.4082483
## [2,] 0.2236068 1.0000000 0.7302967
## [3,] 0.4082483 0.7302967 1.0000000means <- c(0, 0, 0)
set.seed(42)
n <- 1000
x <- rmvnorm(n = n, mean = means, sigma = sigma)
d <- data.frame(x)
p4 <- plot_ly(d, x = ~ X1, y = ~ X2, z = ~ X3,
marker = list(color = ~ X2,
showscale = TRUE)) %>%
add_markers()
p4## identify multivariate outliers
d$mahal <- mahalanobis(d, colMeans(d), cov(d))
d$p_mahal <- pchisq(d$mahal, df=2, lower.tail=FALSE)
d[d$p_mahal < .001, ]## X1 X2 X3 mahal p_mahal
## 274 5.759481 -4.929943 -1.450310 16.41275 0.0002729077
## 330 7.398060 3.344539 4.741212 14.50380 0.0007088271
## 980 -6.295530 -5.365858 -4.367558 14.17879 0.0008339001datenLV$mahal_SSkMa <- mahalanobis(datenLV[,str_subset(string = colnames(datenLV), pattern = "^SSkMa_")], colMeans(datenLV[,str_subset(string = colnames(datenLV), pattern = "^SSkMa_")], na.rm = TRUE), cov(datenLV[,str_subset(string = colnames(datenLV), pattern = "^SSkMa_")], use = "pairwise"))
datenLV$p_mahal_SSkMa <- pchisq(datenLV$mahal_SSkMa, df=3, lower.tail=FALSE)
## identify multivariate outliers
head(datenLV[datenLV$p_mahal_SSkMa < .001 & !is.na(datenLV$p_mahal_SSkMa), c("SSkMa_a", "SSkMa_b", "SSkMa_c", "SSkMa_d", "mahal_SSkMa", "p_mahal_SSkMa")])## SSkMa_a SSkMa_b SSkMa_c SSkMa_d mahal_SSkMa p_mahal_SSkMa
## 8 1 1 4 1 22.80886 4.426227e-05
## 9 1 3 1 4 29.61531 1.662648e-06
## 15 1 1 4 4 17.85799 4.705288e-04
## 25 4 4 1 4 22.55417 5.001386e-05
## 46 4 1 1 4 28.13860 3.396688e-06
## 58 2 4 1 1 17.17129 6.516648e-04sum(datenLV$p_mahal_SSkMa < .001, na.rm = TRUE)## [1] 112datenLV$intravariability_SSkMa <- apply(datenLV[,str_subset(string = colnames(datenLV), pattern = "^SSkMa_")], MARGIN=1, FUN = sd, na.rm=TRUE)
## identify insufficient item responding using variability of answering patterns
head(datenLV[datenLV$intravariability_SSkMa == 0 & !is.na(datenLV$intravariability_SSkMa), c("SSkMa_a", "SSkMa_b", "SSkMa_c", "SSkMa_d", "mahal_SSkMa", "p_mahal_SSkMa")])## SSkMa_a SSkMa_b SSkMa_c SSkMa_d mahal_SSkMa p_mahal_SSkMa
## 4 4 4 4 4 1.3060837 0.727689
## 5 3 3 3 3 0.2810342 0.963555
## 6 4 4 4 4 1.3060837 0.727689
## 7 4 4 4 4 1.3060837 0.727689
## 11 4 4 4 4 1.3060837 0.727689
## 13 3 3 3 3 0.2810342 0.963555sum(datenLV$intravariability_SSkMa == 0, na.rm = TRUE)## [1] 10492 [simulation study: standardized residuals, reliability]
Using the R-Package simstudy it is possible to generate all kinds of data:
I have generated a data set with 3 items (y1-y3) and a data set with 7 items (m1-m7) for different sample sizes. The variables latentvar and errorvar are unknown and for example important in the context of classical test theory as these correspond to the true and error variance):
## varname formula variance dist link
## 1: latentvar 20 0.5 normal identity
## 2: errorvar 4 0.5 normal identity
## 3: y1 latentvar errorvar / 4 normal identity
## 4: y2 latentvar errorvar / 4 normal identity
## 5: y3 latentvar errorvar / 4 normal identity
## 6: m1 latentvar errorvar / 4 normal identity
## 7: m2 latentvar errorvar / 4 normal identity
## 8: m3 latentvar errorvar / 4 normal identity
## 9: m4 latentvar errorvar / 4 normal identity
## 10: m5 latentvar errorvar / 4 normal identity
## 11: m6 latentvar errorvar / 4 normal identity
## 12: m7 latentvar errorvar / 4 normal identityset.seed(111)
dt_50 <- genData(50, def); dt_50 <- as.data.frame(dt_50)
dt_200 <- genData(200, def); dt_200 <- as.data.frame(dt_200)
dt_500 <- genData(500, def); dt_500 <- as.data.frame(dt_500)
dt_100000 <- genData(100000, def); dt_100000 <- as.data.frame(dt_100000)round(x = cor(dt_50[, str_subset(string = colnames(dt_50), pattern = "m")]), digits = 2)## m1 m2 m3 m4 m5 m6 m7
## m1 1.00 0.45 0.39 0.34 0.35 0.43 0.24
## m2 0.45 1.00 0.52 0.34 0.56 0.40 0.49
## m3 0.39 0.52 1.00 0.25 0.44 0.39 0.57
## m4 0.34 0.34 0.25 1.00 0.19 0.30 0.35
## m5 0.35 0.56 0.44 0.19 1.00 0.20 0.37
## m6 0.43 0.40 0.39 0.30 0.20 1.00 0.49
## m7 0.24 0.49 0.57 0.35 0.37 0.49 1.00round(x = cor(dt_100000[, str_subset(string = colnames(dt_100000), pattern = "m")]), digits = 2)## m1 m2 m3 m4 m5 m6 m7
## m1 1.00 0.34 0.33 0.34 0.33 0.33 0.34
## m2 0.34 1.00 0.33 0.34 0.34 0.34 0.34
## m3 0.33 0.33 1.00 0.33 0.33 0.33 0.34
## m4 0.34 0.34 0.33 1.00 0.33 0.34 0.34
## m5 0.33 0.34 0.33 0.33 1.00 0.33 0.33
## m6 0.33 0.34 0.33 0.34 0.33 1.00 0.34
## m7 0.34 0.34 0.34 0.34 0.33 0.34 1.00round(x = cor(dt_50[, str_subset(string = colnames(dt_50), pattern = "m")]), digits = 2) - round(x = cor(dt_100000[, str_subset(string = colnames(dt_100000), pattern = "m")]), digits = 2)## m1 m2 m3 m4 m5 m6 m7
## m1 0.00 0.11 0.06 0.00 0.02 0.10 -0.10
## m2 0.11 0.00 0.19 0.00 0.22 0.06 0.15
## m3 0.06 0.19 0.00 -0.08 0.11 0.06 0.23
## m4 0.00 0.00 -0.08 0.00 -0.14 -0.04 0.01
## m5 0.02 0.22 0.11 -0.14 0.00 -0.13 0.04
## m6 0.10 0.06 0.06 -0.04 -0.13 0.00 0.15
## m7 -0.10 0.15 0.23 0.01 0.04 0.15 0.00sd(dt_50$m1) / sqrt(x = length(dt_50$m1)) ## [1] 0.18367sd(dt_100000$m1) / sqrt(x = length(dt_100000$m1)) ## [1] 0.003876399psych::alpha(cor(dt_50[, str_subset(string = colnames(dt_50), pattern = "m")]))$total## raw_alpha std.alpha G6(smc) average_r S/N median_r
## 0.8132445 0.8132445 0.8146988 0.3835094 4.354593 0.3878958psych::alpha(cor(dt_200[, str_subset(string = colnames(dt_200), pattern = "m")]))$total## raw_alpha std.alpha G6(smc) average_r S/N median_r
## 0.7631399 0.7631399 0.7410928 0.3151959 3.221902 0.3114825psych::alpha(cor(dt_100000[, str_subset(string = colnames(dt_100000), pattern = "m")]))$total## raw_alpha std.alpha G6(smc) average_r S/N median_r
## 0.7781921 0.7781921 0.7504623 0.3338666 3.508406 0.3353357psych::alpha(cor(dt_50[, str_subset(string = colnames(dt_50), pattern = "y")]))$total## raw_alpha std.alpha G6(smc) average_r S/N median_r
## 0.6202657 0.6202657 0.5319161 0.3525301 1.63342 0.3163457psych::alpha(cor(dt_200[, str_subset(string = colnames(dt_200), pattern = "y")]))$total## raw_alpha std.alpha G6(smc) average_r S/N median_r
## 0.5365776 0.5365776 0.4427281 0.2784747 1.157858 0.3205137psych::alpha(cor(dt_100000[, str_subset(string = colnames(dt_100000), pattern = "y")]))$total## raw_alpha std.alpha G6(smc) average_r S/N median_r
## 0.5972012 0.5972012 0.4971001 0.3307499 1.482629 0.3324383psych::omega(m = dt_100000[, str_subset(string = colnames(dt_100000), pattern = "y")], nfactors = 1, plot = FALSE)## Loading required namespace: GPArotation## Omega_h for 1 factor is not meaningful, just omega_t## Warning in schmid(m, nfactors, fm, digits, rotate = rotate, n.obs = n.obs, :
## Omega_h and Omega_asymptotic are not meaningful with one factor## Warning in cov2cor(t(w) %*% r %*% w): diag(.) had 0 or NA entries; non-finite
## result is doubtful## Omega
## Call: omegah(m = m, nfactors = nfactors, fm = fm, key = key, flip = flip,
## digits = digits, title = title, sl = sl, labels = labels,
## plot = plot, n.obs = n.obs, rotate = rotate, Phi = Phi, option = option,
## covar = covar)
## Alpha: 0.6
## G.6: 0.5
## Omega Hierarchical: 0.6
## Omega H asymptotic: 1
## Omega Total 0.6
##
## Schmid Leiman Factor loadings greater than 0.2
## g F1* h2 u2 p2
## y1 0.58 0.34 0.66 1
## y2 0.57 0.33 0.67 1
## y3 0.57 0.33 0.67 1
##
## With eigenvalues of:
## g F1*
## 0.99 0.00
##
## general/max Inf max/min = NaN
## mean percent general = 1 with sd = 0 and cv of 0
## Explained Common Variance of the general factor = 1
##
## The degrees of freedom are 0 and the fit is 0
## The number of observations was 100000 with Chi Square = 0 with prob < NA
## The root mean square of the residuals is 0
## The df corrected root mean square of the residuals is NA
##
## Compare this with the adequacy of just a general factor and no group factors
## The degrees of freedom for just the general factor are 0 and the fit is 0
## The number of observations was 100000 with Chi Square = 0 with prob < NA
## The root mean square of the residuals is 0
## The df corrected root mean square of the residuals is NA
##
## Measures of factor score adequacy
## g F1*
## Correlation of scores with factors 0.77 0
## Multiple R square of scores with factors 0.60 0
## Minimum correlation of factor score estimates 0.19 -1
##
## Total, General and Subset omega for each subset
## g F1*
## Omega total for total scores and subscales 0.6 0.6
## Omega general for total scores and subscales 0.6 0.6
## Omega group for total scores and subscales 0.0 0.0Zusammenfassung:
- die Kovarianzmatrix der Population ist unbekannt und wird durch unsere Daten geschätzt (wenn N gegen unendlich geht, können wir unseren Parameterschätzungen vertrauen -> Konzept des Standardfehlers des Mittelwerts)
- Reliabilität ist abhängig von der Anzahl der Items: unter Zunahme der Anzahl der Items bei gleicher mittlerer Interkorrelation steigt die Reliabilität
- Cronbachs Alpha sollte nur unter Bedingung eines tau-äquivalenten Messmodells und vorheriger Testungs auf Eindimensionalität verwendet werden (Sijtsma (2009)), sonst empfiehlt sich immer modell-basierte Reliabilitätsschätzer wie McDonald’s Omega
3 developing a questionnaire scale
3.1 descreptive analysis using classical test theory
orientiert sich an Buchkapitel 7, 13 in Moosbrugger and Kelava (2020)
\(y_i = \tau_i + \epsilon_i\), aus der Messfehlertheorie folgt die Definition der Reliabilität: \(Rel(Y) = \frac{Var(T)}{Var(T) + Var(E)}\)
\(E(y_i) = E(\tau_i) + E(\epsilon_i)\)
\(E(y_i) = E(\tau_i) + 0\)
über mehrere Items einer Skala lässt sich ein Punktschätzer für den wahren Wert \(\tau_i\) wie folgt berechnen als Summenscore: \(Y = \sum_{i=1}^p y_i\) oder besser interpretierbar als Personmittelwertmittelwert: \(\bar{Y} = \frac{\sum_{i=1}^p y_i}{n}\) ! vorläufige Testwertermittlung (Eindimensionalität, tau-äquivalenten Messmodells muss an sich gegeben sein)
3.1.1 Schwierigkeitsindex
\(P_i = \frac{\sum_{v=1}^n y_{vi}}{n*max(y_i)} *100\)
folgende Zahlen geben die Leichtigkeit des Items an:
datenLV[,str_subset(string = colnames(datenLV), pattern = "^SBezMs_")] <- datenLV[,str_subset(string = colnames(datenLV), pattern = "^SBezMs_")] - 1
datenLV$failitem <- rbinom(n = nrow(datenLV), size = 3, prob = .95)
head(datenLV[, c(str_subset(string = colnames(datenLV), pattern = "^SBezMs_"), "failitem")])## SBezMs_a SBezMs_b SBezMs_c SBezMs_d failitem
## 1 3 2 3 3 2
## 2 2 2 3 3 3
## 3 3 0 3 2 3
## 4 0 3 0 0 3
## 5 3 2 0 0 3
## 6 3 3 3 3 3sum(datenLV$SBezMs_a, na.rm = TRUE) / (sum(!is.na(datenLV$SBezMs_a)) * max(datenLV$SBezMs_a, na.rm = TRUE)) * 100## [1] 78.82201sum(datenLV$SBezMs_b , na.rm = TRUE) / (sum(!is.na(datenLV$SBezMs_b)) * max(datenLV$SBezMs_b, na.rm = TRUE)) * 100## [1] 70.34925sum(datenLV$SBezMs_c , na.rm = TRUE) / (sum(!is.na(datenLV$SBezMs_c)) * max(datenLV$SBezMs_c, na.rm = TRUE)) * 100 ## [1] 78.88418sum(datenLV$SBezMs_d, na.rm = TRUE) / (sum(!is.na(datenLV$SBezMs_d)) * max(datenLV$SBezMs_d, na.rm = TRUE)) * 100## [1] 83.55101sum(datenLV$failitem, na.rm = TRUE) / (sum(!is.na(datenLV$failitem)) * max(datenLV$failitem, na.rm = TRUE)) * 100## [1] 95.50749datenLV[,str_subset(string = colnames(datenLV), pattern = "^SBezMs_")] <- datenLV[,str_subset(string = colnames(datenLV), pattern = "^SBezMs_")] + 13.1.2 Itemvarianz
\(Var(y_i) = \frac{\sum_{v=1}^n (y_{vi} - \bar{y_i})^2}{n}\)
sum((datenLV$SBezMs_a - mean(datenLV$SBezMs_a, na.rm = TRUE))^2, na.rm = TRUE) / sum(!is.na(datenLV$SBezMs_a)) # = var(datenLV$SBezMs_a , na.rm = T)## [1] 0.5393213sum((datenLV$SBezMs_b - mean(datenLV$SBezMs_b, na.rm = TRUE))^2, na.rm = TRUE) / sum(!is.na(datenLV$SBezMs_b))## [1] 0.8131689sum((datenLV$SBezMs_c - mean(datenLV$SBezMs_c, na.rm = TRUE))^2, na.rm = TRUE) / sum(!is.na(datenLV$SBezMs_c)) ## [1] 0.8543597sum((datenLV$SBezMs_d - mean(datenLV$SBezMs_d, na.rm = TRUE))^2, na.rm = TRUE) / sum(!is.na(datenLV$SBezMs_d))## [1] 0.6580054sum((datenLV$failitem - mean(datenLV$failitem, na.rm = TRUE))^2, na.rm = TRUE) / sum(!is.na(datenLV$failitem))## [1] 0.13258443.1.3 Trennschärfe
part-whole korrigierte Trennschärfe \(r_{it(i)}\): \(r_{it(i)} = r_{(y_i, y(i))}\)
cor(datenLV$SBezMs_a, rowSums(datenLV[, c("SBezMs_b", "SBezMs_c", "SBezMs_d")], na.rm = TRUE), use = "complete")## [1] 0.5407678cor(datenLV$SBezMs_b, rowSums(datenLV[, c("SBezMs_a", "SBezMs_c", "SBezMs_d")], na.rm = TRUE), use = "complete")## [1] 0.4083451cor(datenLV$SBezMs_c, rowSums(datenLV[, c("SBezMs_a", "SBezMs_b", "SBezMs_d")], na.rm = TRUE), use = "complete")## [1] 0.4591703cor(datenLV$SBezMs_d, rowSums(datenLV[, c("SBezMs_a", "SBezMs_b", "SBezMs_c")], na.rm = TRUE), use = "complete")## [1] 0.3990633cor(datenLV$failitem, rowSums(datenLV[, c("SBezMs_a", "SBezMs_b", "SBezMs_c", "SBezMs_d")], na.rm = TRUE), use = "complete")## [1] 0.0064537343.1.4 Testwertverteilung
orientiert sich an Buchkapitel 8 in Moosbrugger and Kelava (2020)
## liegt bereits in Daten vor
cor(rowMeans(x = datenLV[,str_subset(string = colnames(datenLV), pattern = "^SBezMs_")]), datenLV$SBezMs, use = "complete")## [1] 1hist(datenLV$SBezMs, freq = FALSE)
abline(v = mean(datenLV$SBezMs, na.rm = TRUE))
lines(density(datenLV$SBezMs[!is.na(datenLV$SBezMs)]), col="red") # empirical density
lines(seq(0, 5, by=.1), dnorm(seq(0, 5, by=.1),
mean(datenLV$SBezMs, na.rm = TRUE), sd(datenLV$SBezMs, na.rm = TRUE)), col="blue") # normal density
sd(x = datenLV$SBezMs, na.rm = TRUE)## [1] 0.6099304moments::skewness(x = datenLV$SBezMs, na.rm = TRUE)## [1] -1.013992moments::kurtosis(x = datenLV$SBezMs, na.rm = TRUE) - 3 # = SPSS output## [1] 0.7847189shapiro.test(x = datenLV$SBezMs)##
## Shapiro-Wilk normality test
##
## data: datenLV$SBezMs
## W = 0.89501, p-value < 2.2e-16für ein normorientierten Vergleich bietet sich eine z-Standardisierung \(\frac{Y_v - \bar{Y}}{SD(Y)}\) an:
datenLV$Zstand_SBezMs <- scale(x = datenLV$SBezMs, center = TRUE, scale = TRUE)
hist(datenLV$Zstand_SBezMs, freq = FALSE)
abline(v = mean(datenLV$Zstand_SBezMs, na.rm = TRUE))
lines(density(datenLV$Zstand_SBezMs[!is.na(datenLV$Zstand_SBezMs)]), col="red")
lines(seq(-4, 4, by=.1), dnorm(seq(-4, 4, by=.1),
mean(datenLV$Zstand_SBezMs, na.rm = TRUE), sd(datenLV$Zstand_SBezMs, na.rm = TRUE)), col="blue")
3.2 exploratory factor analysis
induktive Methode
an sich gehört zur KTT Testung auf Messinvarianz über die klassischen Testmodelle, jedoch muss für diese Eindimensionaltität gegeben sein, hierfür eignet sich eine sogenannte EFA
Verwendung des psych Paketes in R (siehe http://personality-project.org/r/psych/HowTo/factor.pdf), Alternativ eignet sich auch das Statistikprogramm JASP für EFA / CFA: (https://jasp-stats.org/)
einführende Artikel in EFA: Costello and Osborne (2005), Mvududu and Sink (2013) (Anmerkung: es gibt Mischformen zwischen EFA und CFA, wie beispielsweise ESEM: Marsh et al. (2014))
! wichtig es sollte keine principal component analysis gerechnet werden (Relikt der Vergangenheit, Grundprinzipien mit EFA gleich), da hier keine Varianzzerlegung stattfindet.
Ziele der explorativen Faktorenanalyse sind
- die Reduktion der Dimension der Kovarianz- bzw. Korrelationsmatrix
- die Identifizierung von latenten Variablen (z.B. über die Hauptachsenanalyse (Principal Axes Analyses)) und
- die Ausdifferenzierung eines komplexen Merkmalsbereichs in homogene Teilbereiche, d.h. die Variablen werden so gruppiert, dass sie innerhalb der Gruppe möglichst hoch korreliert sind (homogen) und die Gruppen der Variablen zueinander möglichst heterogen sind. Hier wird das gleiche Ziel wie mit einer Clusteranalyse (Latente Klassenanalyse) verfolgt
EFA läuft in vier Schritten ab:
- vorbetrachtende Tests
- die Wahl der Extraktionsmethode
- die Wahl eines Abbruchkriteriums und zuletzt
- die Wahl der Rotationsmethode
zur eigenen Interpretation der Ergebnisse siehe Blog von Michael Clark: https://m-clark.github.io/posts/2020-04-10-psych-explained/
psych::corPlot(r = datenLV[,str_subset(string = colnames(datenLV), pattern = "^SSkMa_|^SSkDe_|^SBezMs_")])
## not accounting for the non-normal / skewed data
efa1 = fa(r = datenLV[,str_subset(string = colnames(datenLV), pattern = "^SSkMa_|^SSkDe_|^SBezMs_")], nfactors = 3, rotate = "oblimin")
fa.diagram(efa1)
efa1## Factor Analysis using method = minres
## Call: fa(r = datenLV[, str_subset(string = colnames(datenLV), pattern = "^SSkMa_|^SSkDe_|^SBezMs_")],
## nfactors = 3, rotate = "oblimin")
## Standardized loadings (pattern matrix) based upon correlation matrix
## MR1 MR3 MR2 h2 u2 com
## SSkDe_a -0.07 0.70 0.04 0.47 0.53 1
## SSkDe_b 0.04 0.51 0.03 0.28 0.72 1
## SSkDe_c 0.09 0.57 -0.01 0.37 0.63 1
## SSkDe_d 0.02 0.75 -0.03 0.56 0.44 1
## SSkMa_a 0.80 -0.02 0.01 0.63 0.37 1
## SSkMa_b 0.53 0.07 0.03 0.32 0.68 1
## SSkMa_c 0.69 0.06 0.00 0.51 0.49 1
## SSkMa_d 0.84 -0.03 -0.01 0.70 0.30 1
## SBezMs_a 0.00 -0.01 0.76 0.57 0.43 1
## SBezMs_b -0.04 0.05 0.53 0.29 0.71 1
## SBezMs_c 0.04 -0.02 0.59 0.35 0.65 1
## SBezMs_d 0.01 0.00 0.53 0.28 0.72 1
##
## MR1 MR3 MR2
## SS loadings 2.16 1.68 1.50
## Proportion Var 0.18 0.14 0.13
## Cumulative Var 0.18 0.32 0.44
## Proportion Explained 0.40 0.31 0.28
## Cumulative Proportion 0.40 0.72 1.00
##
## With factor correlations of
## MR1 MR3 MR2
## MR1 1.00 0.36 0.20
## MR3 0.36 1.00 0.24
## MR2 0.20 0.24 1.00
##
## Mean item complexity = 1
## Test of the hypothesis that 3 factors are sufficient.
##
## The degrees of freedom for the null model are 66 and the objective function was 3.47 with Chi Square of 10398
## The degrees of freedom for the model are 33 and the objective function was 0.43
##
## The root mean square of the residuals (RMSR) is 0.05
## The df corrected root mean square of the residuals is 0.07
##
## The harmonic number of observations is 2811 with the empirical chi square 868.15 with prob < 4.6e-161
## The total number of observations was 3005 with Likelihood Chi Square = 1288.18 with prob < 1.3e-249
##
## Tucker Lewis Index of factoring reliability = 0.757
## RMSEA index = 0.113 and the 90 % confidence intervals are 0.107 0.118
## BIC = 1023.92
## Fit based upon off diagonal values = 0.97
## Measures of factor score adequacy
## MR1 MR3 MR2
## Correlation of (regression) scores with factors 0.92 0.88 0.86
## Multiple R square of scores with factors 0.85 0.77 0.73
## Minimum correlation of possible factor scores 0.70 0.53 0.47### accounting partly for the non-normal / skewed data using choric correlations (limited information approach)
efa2choric <- fa(r = datenLV[,str_subset(string = colnames(datenLV), pattern = "^SSkMa_|^SSkDe_|^SBezMs_")], nfactors = 3, rotate = "oblimin", fm = "wls", max.iter = 500, cor = "poly", scores = "Bartlett")
efa2choric## Factor Analysis using method = wls
## Call: fa(r = datenLV[, str_subset(string = colnames(datenLV), pattern = "^SSkMa_|^SSkDe_|^SBezMs_")],
## nfactors = 3, rotate = "oblimin", scores = "Bartlett", max.iter = 500,
## fm = "wls", cor = "poly")
## Standardized loadings (pattern matrix) based upon correlation matrix
## WLS1 WLS3 WLS2 h2 u2 com
## SSkDe_a -0.09 0.76 0.04 0.55 0.45 1.0
## SSkDe_b 0.04 0.57 0.03 0.36 0.64 1.0
## SSkDe_c 0.10 0.66 -0.01 0.49 0.51 1.1
## SSkDe_d 0.03 0.82 -0.03 0.67 0.33 1.0
## SSkMa_a 0.86 -0.03 0.01 0.72 0.28 1.0
## SSkMa_b 0.62 0.06 0.02 0.43 0.57 1.0
## SSkMa_c 0.77 0.07 0.00 0.64 0.36 1.0
## SSkMa_d 0.90 -0.02 0.00 0.80 0.20 1.0
## SBezMs_a 0.00 -0.01 0.83 0.68 0.32 1.0
## SBezMs_b -0.05 0.05 0.59 0.36 0.64 1.0
## SBezMs_c 0.04 -0.02 0.69 0.48 0.52 1.0
## SBezMs_d 0.00 0.01 0.63 0.40 0.60 1.0
##
## WLS1 WLS3 WLS2
## SS loadings 2.60 2.06 1.92
## Proportion Var 0.22 0.17 0.16
## Cumulative Var 0.22 0.39 0.55
## Proportion Explained 0.39 0.31 0.29
## Cumulative Proportion 0.39 0.71 1.00
##
## With factor correlations of
## WLS1 WLS3 WLS2
## WLS1 1.00 0.41 0.23
## WLS3 0.41 1.00 0.27
## WLS2 0.23 0.27 1.00
##
## Mean item complexity = 1
## Test of the hypothesis that 3 factors are sufficient.
##
## The degrees of freedom for the null model are 66 and the objective function was 5.63 with Chi Square of 16873.08
## The degrees of freedom for the model are 33 and the objective function was 0.89
##
## The root mean square of the residuals (RMSR) is 0.06
## The df corrected root mean square of the residuals is 0.08
##
## The harmonic number of observations is 2811 with the empirical chi square 1148.38 with prob < 5e-220
## The total number of observations was 3005 with Likelihood Chi Square = 2678.78 with prob < 0
##
## Tucker Lewis Index of factoring reliability = 0.685
## RMSEA index = 0.163 and the 90 % confidence intervals are 0.158 0.169
## BIC = 2414.51
## Fit based upon off diagonal values = 0.97
## Measures of factor score adequacy
## WLS1 WLS3 WLS2
## Correlation of (regression) scores with factors 0.95 0.91 0.90
## Multiple R square of scores with factors 0.91 0.84 0.81
## Minimum correlation of possible factor scores 0.81 0.67 0.62### model based reliability score
omega(datenLV[,str_subset(string = colnames(datenLV), pattern = "^SSkMa_|^SSkDe_|^SBezMs_")])
## Omega
## Call: omegah(m = m, nfactors = nfactors, fm = fm, key = key, flip = flip,
## digits = digits, title = title, sl = sl, labels = labels,
## plot = plot, n.obs = n.obs, rotate = rotate, Phi = Phi, option = option,
## covar = covar)
## Alpha: 0.76
## G.6: 0.81
## Omega Hierarchical: 0.45
## Omega H asymptotic: 0.54
## Omega Total 0.83
##
## Schmid Leiman Factor loadings greater than 0.2
## g F1* F2* F3* h2 u2 p2
## SSkDe_a 0.43 0.53 0.47 0.53 0.40
## SSkDe_b 0.37 0.38 0.28 0.72 0.48
## SSkDe_c 0.42 0.43 0.37 0.63 0.48
## SSkDe_d 0.49 0.56 0.56 0.44 0.43
## SSkMa_a 0.43 0.67 0.63 0.37 0.29
## SSkMa_b 0.35 0.44 0.32 0.68 0.38
## SSkMa_c 0.42 0.57 0.51 0.49 0.35
## SSkMa_d 0.45 0.70 0.70 0.30 0.28
## SBezMs_a 0.27 0.71 0.57 0.43 0.13
## SBezMs_b 0.20 0.50 0.29 0.71 0.14
## SBezMs_c 0.22 0.55 0.35 0.65 0.14
## SBezMs_d 0.20 0.49 0.28 0.72 0.14
##
## With eigenvalues of:
## g F1* F2* F3*
## 1.63 1.48 0.93 1.30
##
## general/max 1.1 max/min = 1.59
## mean percent general = 0.3 with sd = 0.14 and cv of 0.45
## Explained Common Variance of the general factor = 0.31
##
## The degrees of freedom are 33 and the fit is 0.43
## The number of observations was 3005 with Chi Square = 1288.18 with prob < 1.3e-249
## The root mean square of the residuals is 0.05
## The df corrected root mean square of the residuals is 0.07
## RMSEA index = 0.113 and the 10 % confidence intervals are 0.107 0.118
## BIC = 1023.92
##
## Compare this with the adequacy of just a general factor and no group factors
## The degrees of freedom for just the general factor are 54 and the fit is 2.02
## The number of observations was 3005 with Chi Square = 6062.27 with prob < 0
## The root mean square of the residuals is 0.17
## The df corrected root mean square of the residuals is 0.19
##
## RMSEA index = 0.192 and the 10 % confidence intervals are 0.188 0.197
## BIC = 5629.84
##
## Measures of factor score adequacy
## g F1* F2* F3*
## Correlation of scores with factors 0.70 0.81 0.69 0.81
## Multiple R square of scores with factors 0.49 0.65 0.48 0.66
## Minimum correlation of factor score estimates -0.03 0.30 -0.05 0.32
##
## Total, General and Subset omega for each subset
## g F1* F2* F3*
## Omega total for total scores and subscales 0.83 0.82 0.74 0.70
## Omega general for total scores and subscales 0.45 0.26 0.33 0.10
## Omega group for total scores and subscales 0.36 0.56 0.41 0.61Wenn die Anzahl der zu bestimmenden Faktoren unklar ist bietet sich die Verwendung von Scree plots an:
efa3 <- fa.parallel(x = datenLV[,str_subset(string = colnames(datenLV), pattern = "^SSkMa_|^SSkDe_|^SBezMs_")], fa = "fa",n.iter=50)
## Parallel analysis suggests that the number of factors = 4 and the number of components = NAefa3## Call: fa.parallel(x = datenLV[, str_subset(string = colnames(datenLV),
## pattern = "^SSkMa_|^SSkDe_|^SBezMs_")], fa = "fa", n.iter = 50)
## Parallel analysis suggests that the number of factors = 4 and the number of components = NA
##
## Eigen Values of
##
## eigen values of factors
## [1] 2.73 1.05 0.83 0.29 -0.10 -0.15 -0.20 -0.22 -0.31 -0.36 -0.42 -0.42
##
## eigen values of simulated factors
## [1] 0.31 0.08 0.06 0.05 0.03 0.02 0.00 -0.01 -0.03 -0.05 -0.06 -0.09
##
## eigen values of components
## [1] 3.43 1.89 1.60 1.04 0.78 0.62 0.61 0.49 0.47 0.45 0.33 0.28
##
## eigen values of simulated components
## [1] NA3.2.1 classical test models / measurment invariance
Im Folgenden wollen wir die Items zu dem Matheselbstkonzept genauer analysieren - ohne die Testung von tau-äquivalentem Modell, sowie Eindimensionalität berechnen wir vorläufig nur McDonald’s Omega:
psych::omega(datenLV[,str_subset(string = colnames(datenLV), pattern = "^SSkMa_")], nfactors = 1)## Omega_h for 1 factor is not meaningful, just omega_t## Warning in schmid(m, nfactors, fm, digits, rotate = rotate, n.obs = n.obs, :
## Omega_h and Omega_asymptotic are not meaningful with one factor## Omega
## Call: omegah(m = m, nfactors = nfactors, fm = fm, key = key, flip = flip,
## digits = digits, title = title, sl = sl, labels = labels,
## plot = plot, n.obs = n.obs, rotate = rotate, Phi = Phi, option = option,
## covar = covar)
## Alpha: 0.81
## G.6: 0.78
## Omega Hierarchical: 0.82
## Omega H asymptotic: 1
## Omega Total 0.82
##
## Schmid Leiman Factor loadings greater than 0.2
## g F1* h2 u2 p2
## SSkMa_a 0.81 0.65 0.35 1
## SSkMa_b 0.56 0.31 0.69 1
## SSkMa_c 0.70 0.49 0.51 1
## SSkMa_d 0.83 0.69 0.31 1
##
## With eigenvalues of:
## g F1*
## 2.1 0.0
##
## general/max 3.862933e+16 max/min = 1
## mean percent general = 1 with sd = 0 and cv of 0
## Explained Common Variance of the general factor = 1
##
## The degrees of freedom are 2 and the fit is 0.05
## The number of observations was 3005 with Chi Square = 143.45 with prob < 7.1e-32
## The root mean square of the residuals is 0.04
## The df corrected root mean square of the residuals is 0.07
## RMSEA index = 0.153 and the 10 % confidence intervals are 0.133 0.175
## BIC = 127.43
##
## Compare this with the adequacy of just a general factor and no group factors
## The degrees of freedom for just the general factor are 2 and the fit is 0.05
## The number of observations was 3005 with Chi Square = 143.45 with prob < 7.1e-32
## The root mean square of the residuals is 0.04
## The df corrected root mean square of the residuals is 0.07
##
## RMSEA index = 0.153 and the 10 % confidence intervals are 0.133 0.175
## BIC = 127.43
##
## Measures of factor score adequacy
## g F1*
## Correlation of scores with factors 0.92 0
## Multiple R square of scores with factors 0.85 0
## Minimum correlation of factor score estimates 0.69 -1
##
## Total, General and Subset omega for each subset
## g F1*
## Omega total for total scores and subscales 0.82 0.82
## Omega general for total scores and subscales 0.82 0.82
## Omega group for total scores and subscales 0.00 0.00Um auf Messinvarianz zu testen, müssen wir das Messmodell über eine sogenannte Modellsyntax eingeben, um darauf folgend das R Paket lavaan verwenden zu können:
Achtung: CFAs werden geschätzt mittels maximum likelihood (ML), weiter unten in Abschnitt CFA / SEM für die Daten besser geeignete Schätzmethode (jedoch ist ML hier zielführend da hiermit über den likelihood ratio test ein Modellvergleich gerechnet werden kann):
classical test models (Voraussetzung Reliabilitätsanalysen, Sparsamkeit des Modells; gleiches Prinzip wie Messinvarianz weiter unten)
cong.model <- '
SSmath =~ lam1*SSkMa_a + lam2*SSkMa_b + lam3*SSkMa_c + lam4*SSkMa_d
SSkMa_a ~~ var1*SSkMa_a
SSkMa_b ~~ var2*SSkMa_b
SSkMa_c ~~ var3*SSkMa_c
SSkMa_d ~~ var4*SSkMa_d
SSkMa_a ~ mean1*1
SSkMa_b ~ mean2*1
SSkMa_c ~ mean3*1
SSkMa_d ~ mean4*1
'
# identification: Fixed factor
cong.fit <-sem(cong.model, data = datenLV, std.lv = TRUE)
summary(cong.fit, standardized = TRUE, fit.measures=TRUE)## lavaan 0.6-8 ended normally after 16 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 12
##
## Used Total
## Number of observations 2843 3005
##
## Model Test User Model:
##
## Test statistic 124.430
## Degrees of freedom 2
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 4111.594
## Degrees of freedom 6
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.970
## Tucker-Lewis Index (TLI) 0.911
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -12967.449
## Loglikelihood unrestricted model (H1) -12905.234
##
## Akaike (AIC) 25958.898
## Bayesian (BIC) 26030.329
## Sample-size adjusted Bayesian (BIC) 25992.201
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.147
## 90 Percent confidence interval - lower 0.125
## 90 Percent confidence interval - upper 0.169
## P-value RMSEA <= 0.05 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.028
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## SSmath =~
## SSkMa_a (lam1) 0.733 0.016 45.845 0.000 0.733 0.786
## SSkMa_b (lam2) 0.603 0.020 29.672 0.000 0.603 0.554
## SSkMa_c (lam3) 0.582 0.014 40.830 0.000 0.582 0.717
## SSkMa_d (lam4) 0.674 0.013 49.973 0.000 0.674 0.841
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .SSkMa_a (men1) 3.177 0.017 181.586 0.000 3.177 3.406
## .SSkMa_b (men2) 2.943 0.020 144.193 0.000 2.943 2.704
## .SSkMa_c (men3) 3.319 0.015 217.876 0.000 3.319 4.086
## .SSkMa_d (men4) 3.337 0.015 221.820 0.000 3.337 4.160
## SSmath 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .SSkMa_a (var1) 0.332 0.013 25.253 0.000 0.332 0.382
## .SSkMa_b (var2) 0.820 0.024 34.594 0.000 0.820 0.693
## .SSkMa_c (var3) 0.320 0.011 29.946 0.000 0.320 0.486
## .SSkMa_d (var4) 0.189 0.009 19.857 0.000 0.189 0.293
## SSmath 1.000 1.000 1.000semPlot::semPaths(object = cong.fit, what = "est")
tauequi.model <- '
SSmath =~ lam1*SSkMa_a + lam2*SSkMa_b + lam3*SSkMa_c + lam4*SSkMa_d
SSkMa_a ~~ var1*SSkMa_a
SSkMa_b ~~ var2*SSkMa_b
SSkMa_c ~~ var3*SSkMa_c
SSkMa_d ~~ var4*SSkMa_d
SSkMa_a ~ mean1*1
SSkMa_b ~ mean2*1
SSkMa_c ~ mean3*1
SSkMa_d ~ mean4*1
# fix variance of SSmath factor
SSmath ~~ 1*SSmath
# constraints
lam1 == lam2
lam2 == lam3
lam3 == lam4
'
# identification: Fixed factor
tauequi.fit <-sem(tauequi.model, data = datenLV, std.lv = TRUE)
summary(tauequi.fit, standardized = TRUE, fit.measures=TRUE)## lavaan 0.6-8 ended normally after 12 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 12
## Number of equality constraints 3
##
## Used Total
## Number of observations 2843 3005
##
## Model Test User Model:
##
## Test statistic 209.808
## Degrees of freedom 5
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 4111.594
## Degrees of freedom 6
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.950
## Tucker-Lewis Index (TLI) 0.940
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -13010.138
## Loglikelihood unrestricted model (H1) -12905.234
##
## Akaike (AIC) 26038.276
## Bayesian (BIC) 26091.849
## Sample-size adjusted Bayesian (BIC) 26063.253
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.120
## 90 Percent confidence interval - lower 0.106
## 90 Percent confidence interval - upper 0.134
## P-value RMSEA <= 0.05 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.065
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## SSmath =~
## SSkMa_a (lam1) 0.655 0.010 63.090 0.000 0.655 0.734
## SSkMa_b (lam2) 0.655 0.010 63.090 0.000 0.655 0.586
## SSkMa_c (lam3) 0.655 0.010 63.090 0.000 0.655 0.767
## SSkMa_d (lam4) 0.655 0.010 63.090 0.000 0.655 0.830
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .SSkMa_a (men1) 3.177 0.017 189.961 0.000 3.177 3.563
## .SSkMa_b (men2) 2.943 0.021 140.437 0.000 2.943 2.634
## .SSkMa_c (men3) 3.319 0.016 207.225 0.000 3.319 3.886
## .SSkMa_d (men4) 3.337 0.015 225.422 0.000 3.337 4.228
## SSmath 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .SSkMa_a (var1) 0.366 0.012 30.452 0.000 0.366 0.461
## .SSkMa_b (var2) 0.820 0.024 34.536 0.000 0.820 0.657
## .SSkMa_c (var3) 0.300 0.010 28.781 0.000 0.300 0.412
## .SSkMa_d (var4) 0.194 0.008 23.975 0.000 0.194 0.312
## SSmath 1.000 1.000 1.000
##
## Constraints:
## |Slack|
## lam1 - (lam2) 0.000
## lam2 - (lam3) 0.000
## lam3 - (lam4) 0.000anova(cong.fit, tauequi.fit) # LRT## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## cong.fit 2 25959 26030 124.43
## tauequi.fit 5 26038 26092 209.81 85.378 3 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1fit.stats <- rbind(fitmeasures(cong.fit, fit.measures = c("chisq", "df", "rmsea", "tli", "cfi", "aic")),
fitmeasures(tauequi.fit, fit.measures = c("chisq", "df", "rmsea", "tli", "cfi", "aic")))
rownames(fit.stats) <- c("configural", "weak invariance")
fit.stats## chisq df rmsea tli cfi aic
## configural 124.4304 2 0.1467375 0.9105389 0.9701796 25958.90
## weak invariance 209.8084 5 0.1200330 0.9401377 0.9501148 26038.28parallel.model <- '
SSmath =~ lam1*SSkMa_a + lam2*SSkMa_b + lam3*SSkMa_c + lam4*SSkMa_d
SSkMa_a ~~ var1*SSkMa_a
SSkMa_b ~~ var2*SSkMa_b
SSkMa_c ~~ var3*SSkMa_c
SSkMa_d ~~ var4*SSkMa_d
SSkMa_a ~ mean1*1
SSkMa_b ~ mean2*1
SSkMa_c ~ mean3*1
SSkMa_d ~ mean4*1
# fix variance of SSmath factor
SSmath ~~ 1*SSmath
# constraints
lam1 == lam2
lam2 == lam3
lam3 == lam4
var1 == var2
var2 == var3
var3 == var4
'
# identification: Fixed factor
parallel.fit <-sem(parallel.model, data = datenLV, std.lv = TRUE)
summary(parallel.fit, standardized = TRUE, fit.measures=TRUE)## lavaan 0.6-8 ended normally after 4 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 12
## Number of equality constraints 6
##
## Used Total
## Number of observations 2843 3005
##
## Model Test User Model:
##
## Test statistic 1157.170
## Degrees of freedom 8
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 4111.594
## Degrees of freedom 6
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.720
## Tucker-Lewis Index (TLI) 0.790
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -13483.819
## Loglikelihood unrestricted model (H1) -12905.234
##
## Akaike (AIC) 26979.637
## Bayesian (BIC) 27015.353
## Sample-size adjusted Bayesian (BIC) 26996.289
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.225
## 90 Percent confidence interval - lower 0.214
## 90 Percent confidence interval - upper 0.236
## P-value RMSEA <= 0.05 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.144
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## SSmath =~
## SSkMa_a (lam1) 0.648 0.011 59.955 0.000 0.648 0.708
## SSkMa_b (lam2) 0.648 0.011 59.955 0.000 0.648 0.708
## SSkMa_c (lam3) 0.648 0.011 59.955 0.000 0.648 0.708
## SSkMa_d (lam4) 0.648 0.011 59.955 0.000 0.648 0.708
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .SSkMa_a (men1) 3.177 0.017 184.882 0.000 3.177 3.467
## .SSkMa_b (men2) 2.943 0.017 171.290 0.000 2.943 3.213
## .SSkMa_c (men3) 3.319 0.017 193.132 0.000 3.319 3.622
## .SSkMa_d (men4) 3.337 0.017 194.196 0.000 3.337 3.642
## SSmath 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .SSkMa_a (var1) 0.419 0.006 65.303 0.000 0.419 0.499
## .SSkMa_b (var2) 0.419 0.006 65.303 0.000 0.419 0.499
## .SSkMa_c (var3) 0.419 0.006 65.303 0.000 0.419 0.499
## .SSkMa_d (var4) 0.419 0.006 65.303 0.000 0.419 0.499
## SSmath 1.000 1.000 1.000
##
## Constraints:
## |Slack|
## lam1 - (lam2) 0.000
## lam2 - (lam3) 0.000
## lam3 - (lam4) 0.000
## var1 - (var2) 0.000
## var2 - (var3) 0.000
## var3 - (var4) 0.000anova(cong.fit, tauequi.fit, parallel.fit) # LRT## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## cong.fit 2 25959 26030 124.43
## tauequi.fit 5 26038 26092 209.81 85.38 3 < 2.2e-16 ***
## parallel.fit 8 26980 27015 1157.17 947.36 3 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1semPlot::semPaths(object = parallel.fit, what = "est")
measurment invariance (longitudinal data, multi-group analysis)
zur eigenen Interpretation der Ergebnisse siehe: https://rstudio-pubs-static.s3.amazonaws.com/194879_192b64ad567743d392b559d650b95a3b.html
CFAmodel <- ' SSmath =~ SSkMa_a + SSkMa_b + SSkMa_c + SSkMa_d'
fit <- cfa(CFAmodel, data=datenLV) # ! ML
summary(fit, fit.measures=TRUE)## lavaan 0.6-8 ended normally after 19 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 8
##
## Used Total
## Number of observations 2843 3005
##
## Model Test User Model:
##
## Test statistic 124.430
## Degrees of freedom 2
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 4111.594
## Degrees of freedom 6
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.970
## Tucker-Lewis Index (TLI) 0.911
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -12967.449
## Loglikelihood unrestricted model (H1) -12905.234
##
## Akaike (AIC) 25950.898
## Bayesian (BIC) 25998.519
## Sample-size adjusted Bayesian (BIC) 25973.100
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.147
## 90 Percent confidence interval - lower 0.125
## 90 Percent confidence interval - upper 0.169
## P-value RMSEA <= 0.05 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.033
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## SSmath =~
## SSkMa_a 1.000
## SSkMa_b 0.823 0.029 28.041 0.000
## SSkMa_c 0.794 0.022 36.756 0.000
## SSkMa_d 0.920 0.023 40.728 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .SSkMa_a 0.332 0.013 25.253 0.000
## .SSkMa_b 0.820 0.024 34.594 0.000
## .SSkMa_c 0.320 0.011 29.946 0.000
## .SSkMa_d 0.189 0.009 19.857 0.000
## SSmath 0.538 0.023 22.923 0.000table(datenLV$Emigr)##
## Mig keinMig
## 493 1966configural <- cfa(CFAmodel, data=datenLV, group = "Emigr")## Warning in lav_data_full(data = data, group = group, cluster = cluster, : lavaan WARNING: group variable 'Emigr' contains missing valuessummary(configural, fit.measures=TRUE)## lavaan 0.6-8 ended normally after 35 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 24
##
## Number of observations per group: Used Total
## keinMig 1878 1966
## Mig 465 493
##
## Model Test User Model:
##
## Test statistic 98.189
## Degrees of freedom 4
## P-value (Chi-square) 0.000
## Test statistic for each group:
## keinMig 72.077
## Mig 26.112
##
## Model Test Baseline Model:
##
## Test statistic 3389.720
## Degrees of freedom 12
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.972
## Tucker-Lewis Index (TLI) 0.916
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -10511.368
## Loglikelihood unrestricted model (H1) -10462.274
##
## Akaike (AIC) 21070.736
## Bayesian (BIC) 21208.957
## Sample-size adjusted Bayesian (BIC) 21132.704
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.142
## 90 Percent confidence interval - lower 0.118
## 90 Percent confidence interval - upper 0.167
## P-value RMSEA <= 0.05 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.027
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [keinMig]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## SSmath =~
## SSkMa_a 1.000
## SSkMa_b 0.865 0.037 23.226 0.000
## SSkMa_c 0.814 0.027 29.652 0.000
## SSkMa_d 0.931 0.028 32.753 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .SSkMa_a 3.238 0.021 155.739 0.000
## .SSkMa_b 3.013 0.025 122.777 0.000
## .SSkMa_c 3.356 0.018 184.052 0.000
## .SSkMa_d 3.377 0.018 189.921 0.000
## SSmath 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .SSkMa_a 0.324 0.015 21.236 0.000
## .SSkMa_b 0.766 0.027 27.915 0.000
## .SSkMa_c 0.301 0.012 24.207 0.000
## .SSkMa_d 0.171 0.011 15.854 0.000
## SSmath 0.488 0.027 18.289 0.000
##
##
## Group 2 [Mig]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## SSmath =~
## SSkMa_a 1.000
## SSkMa_b 0.758 0.066 11.542 0.000
## SSkMa_c 0.699 0.050 14.064 0.000
## SSkMa_d 0.916 0.053 17.127 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .SSkMa_a 3.069 0.044 69.482 0.000
## .SSkMa_b 2.798 0.051 55.224 0.000
## .SSkMa_c 3.239 0.039 82.839 0.000
## .SSkMa_d 3.241 0.040 80.897 0.000
## SSmath 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .SSkMa_a 0.280 0.033 8.428 0.000
## .SSkMa_b 0.833 0.059 14.087 0.000
## .SSkMa_c 0.404 0.031 13.177 0.000
## .SSkMa_d 0.219 0.027 8.049 0.000
## SSmath 0.628 0.063 9.963 0.000weak.invariance <- cfa(CFAmodel, data=datenLV, group = "Emigr", group.equal = "loadings")## Warning in lav_data_full(data = data, group = group, cluster = cluster, : lavaan WARNING: group variable 'Emigr' contains missing valuessummary(weak.invariance, fit.measures = TRUE)## lavaan 0.6-8 ended normally after 27 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 24
## Number of equality constraints 3
##
## Number of observations per group: Used Total
## keinMig 1878 1966
## Mig 465 493
##
## Model Test User Model:
##
## Test statistic 104.031
## Degrees of freedom 7
## P-value (Chi-square) 0.000
## Test statistic for each group:
## keinMig 73.273
## Mig 30.758
##
## Model Test Baseline Model:
##
## Test statistic 3389.720
## Degrees of freedom 12
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.971
## Tucker-Lewis Index (TLI) 0.951
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -10514.289
## Loglikelihood unrestricted model (H1) -10462.274
##
## Akaike (AIC) 21070.578
## Bayesian (BIC) 21191.521
## Sample-size adjusted Bayesian (BIC) 21124.800
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.109
## 90 Percent confidence interval - lower 0.091
## 90 Percent confidence interval - upper 0.128
## P-value RMSEA <= 0.05 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.032
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [keinMig]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## SSmath =~
## SSkMa_a 1.000
## SSkMa_b (.p2.) 0.840 0.032 25.944 0.000
## SSkMa_c (.p3.) 0.790 0.024 32.950 0.000
## SSkMa_d (.p4.) 0.927 0.025 37.016 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .SSkMa_a 3.238 0.021 154.981 0.000
## .SSkMa_b 3.013 0.024 123.393 0.000
## .SSkMa_c 3.356 0.018 185.466 0.000
## .SSkMa_d 3.377 0.018 189.351 0.000
## SSmath 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .SSkMa_a 0.322 0.015 21.310 0.000
## .SSkMa_b 0.768 0.027 28.123 0.000
## .SSkMa_c 0.304 0.012 24.751 0.000
## .SSkMa_d 0.169 0.011 16.002 0.000
## SSmath 0.498 0.026 19.451 0.000
##
##
## Group 2 [Mig]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## SSmath =~
## SSkMa_a 1.000
## SSkMa_b (.p2.) 0.840 0.032 25.944 0.000
## SSkMa_c (.p3.) 0.790 0.024 32.950 0.000
## SSkMa_d (.p4.) 0.927 0.025 37.016 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .SSkMa_a 3.069 0.043 70.625 0.000
## .SSkMa_b 2.798 0.052 54.123 0.000
## .SSkMa_c 3.239 0.040 80.088 0.000
## .SSkMa_d 3.241 0.040 81.950 0.000
## SSmath 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .SSkMa_a 0.292 0.029 10.086 0.000
## .SSkMa_b 0.829 0.059 14.044 0.000
## .SSkMa_c 0.395 0.030 12.983 0.000
## .SSkMa_d 0.223 0.024 9.499 0.000
## SSmath 0.586 0.049 11.977 0.000anova(weak.invariance, configural) # LRT## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## configural 4 21071 21209 98.189
## weak.invariance 7 21071 21192 104.031 5.8415 3 0.1196fit.stats <- rbind(fitmeasures(configural, fit.measures = c("chisq", "df", "rmsea", "tli", "cfi", "aic")),
fitmeasures(weak.invariance, fit.measures = c("chisq", "df", "rmsea", "tli", "cfi", "aic")))
rownames(fit.stats) <- c("configural", "weak invariance")
fit.stats## chisq df rmsea tli cfi aic
## configural 98.18934 4 0.1417750 0.9163436 0.9721145 21070.74
## weak invariance 104.03081 7 0.1087764 0.9507542 0.9712733 21070.58strong.invariance <- cfa(CFAmodel, data=datenLV, group = "Emigr", group.equal = c( "loadings", "intercepts"))## Warning in lav_data_full(data = data, group = group, cluster = cluster, : lavaan WARNING: group variable 'Emigr' contains missing valuessummary(strong.invariance, fit.measures = TRUE)## lavaan 0.6-8 ended normally after 39 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 25
## Number of equality constraints 7
##
## Number of observations per group: Used Total
## keinMig 1878 1966
## Mig 465 493
##
## Model Test User Model:
##
## Test statistic 107.336
## Degrees of freedom 10
## P-value (Chi-square) 0.000
## Test statistic for each group:
## keinMig 73.780
## Mig 33.556
##
## Model Test Baseline Model:
##
## Test statistic 3389.720
## Degrees of freedom 12
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.971
## Tucker-Lewis Index (TLI) 0.965
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -10515.941
## Loglikelihood unrestricted model (H1) -10462.274
##
## Akaike (AIC) 21067.883
## Bayesian (BIC) 21171.548
## Sample-size adjusted Bayesian (BIC) 21114.359
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.091
## 90 Percent confidence interval - lower 0.076
## 90 Percent confidence interval - upper 0.107
## P-value RMSEA <= 0.05 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.032
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
##
## Group 1 [keinMig]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## SSmath =~
## SSkMa_a 1.000
## SSkMa_b (.p2.) 0.844 0.032 26.189 0.000
## SSkMa_c (.p3.) 0.788 0.024 33.102 0.000
## SSkMa_d (.p4.) 0.925 0.025 37.264 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .SSkMa_a (.10.) 3.237 0.020 159.328 0.000
## .SSkMa_b (.11.) 2.999 0.023 130.339 0.000
## .SSkMa_c (.12.) 3.358 0.017 192.037 0.000
## .SSkMa_d (.13.) 3.380 0.018 192.150 0.000
## SSmath 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .SSkMa_a 0.321 0.015 21.285 0.000
## .SSkMa_b 0.767 0.027 28.090 0.000
## .SSkMa_c 0.304 0.012 24.775 0.000
## .SSkMa_d 0.170 0.011 16.109 0.000
## SSmath 0.499 0.026 19.521 0.000
##
##
## Group 2 [Mig]:
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## SSmath =~
## SSkMa_a 1.000
## SSkMa_b (.p2.) 0.844 0.032 26.189 0.000
## SSkMa_c (.p3.) 0.788 0.024 33.102 0.000
## SSkMa_d (.p4.) 0.925 0.025 37.264 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .SSkMa_a (.10.) 3.237 0.020 159.328 0.000
## .SSkMa_b (.11.) 2.999 0.023 130.339 0.000
## .SSkMa_c (.12.) 3.358 0.017 192.037 0.000
## .SSkMa_d (.13.) 3.380 0.018 192.150 0.000
## SSmath -0.164 0.042 -3.867 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .SSkMa_a 0.291 0.029 10.062 0.000
## .SSkMa_b 0.832 0.059 14.035 0.000
## .SSkMa_c 0.395 0.030 12.989 0.000
## .SSkMa_d 0.224 0.024 9.543 0.000
## SSmath 0.587 0.049 11.990 0.000anova(strong.invariance, weak.invariance, configural)## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## configural 4 21071 21209 98.189
## weak.invariance 7 21071 21192 104.031 5.8415 3 0.1196
## strong.invariance 10 21068 21172 107.336 3.3049 3 0.3470strict.invariance <- cfa(CFAmodel, data=datenLV, group = "Emigr", group.equal = c( "loadings", "intercepts", "residuals"))## Warning in lav_data_full(data = data, group = group, cluster = cluster, : lavaan WARNING: group variable 'Emigr' contains missing valuesanova(strict.invariance, strong.invariance, weak.invariance, configural)## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## configural 4 21071 21209 98.189
## weak.invariance 7 21071 21192 104.031 5.8415 3 0.119583
## strong.invariance 10 21068 21172 107.336 3.3049 3 0.346958
## strict.invariance 14 21077 21158 124.388 17.0527 4 0.001888 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 13.3 confirmatory factor analysis
Verwendung des lavaan Paketes in R (siehe https://lavaan.ugent.be/), google group lavaan (https://groups.google.com/g/lavaan); es empfiehlt sich jedoch für komplexe Analysen Mplus zu verwenden (FIML, Bayesian SEM, …)
wichtigstes Grundlagenbuch zu SEM: Bollen (1989)
3.3.1 short theoretical: focus the structural modelling perspective
deduktive Methode
Context-Process-Input-Output model (bekannt in deutschsprachigen Raum durch Ditton (2000), entwickelt von Stufflebeam (1971); klare Ausführungen in Keller (2014)):
Dies lässt sich zusammenbauen zu einem nomologischen Netzwerk (= Testung Konstruktvalidität):
Welche möglicherweise Variablen interessant sind lässt sich aus einer graphischen theoretischen Ausarbeitung (Pfaddiagramm) schrittweise aufbauen (Kapitel 7 “causal models” in Jaccard and Jacoby (2020)):
3.3.2 CFA (measurement model)
Abschnitt CFA / SEM orientiert sich an Kapitel 9-12 in Hair et al. (2019):
an sich sollten die einzelnen Messmodelle (CFAs) getrennt berechnet werden, hier wird jedoch aus Zeitdrücken direkt eine CFA erster Ordnung (first order CFA) für alle Messmodelle, die im Strukturgleichungsmodell verwendet werden gerechnet:
firstorderCFA <- '
SSmath =~ SSkMa_a + SSkMa_b + SSkMa_c + SSkMa_d
SSgerman =~ SSkDe_a + SSkDe_b + SSkDe_c + SSkDe_d
SozInt =~ SBezMs_a + SBezMs_b + SBezMs_c + SBezMs_d
Abilities =~ wle_lesen + wle_hoeren + wle_mathe
'
# identification: Fixed factor
fit <-sem(firstorderCFA, data = datenLV, std.lv = TRUE)
summary(fit, standardized = TRUE, fit.measures=TRUE)## lavaan 0.6-8 ended normally after 24 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 36
##
## Used Total
## Number of observations 2566 3005
##
## Model Test User Model:
##
## Test statistic 1514.039
## Degrees of freedom 84
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 11646.307
## Degrees of freedom 105
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.876
## Tucker-Lewis Index (TLI) 0.845
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -45769.003
## Loglikelihood unrestricted model (H1) -45011.984
##
## Akaike (AIC) 91610.007
## Bayesian (BIC) 91820.610
## Sample-size adjusted Bayesian (BIC) 91706.228
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.081
## 90 Percent confidence interval - lower 0.078
## 90 Percent confidence interval - upper 0.085
## P-value RMSEA <= 0.05 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.054
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## SSmath =~
## SSkMa_a 0.733 0.017 44.281 0.000 0.733 0.789
## SSkMa_b 0.618 0.021 29.186 0.000 0.618 0.569
## SSkMa_c 0.584 0.015 39.403 0.000 0.584 0.722
## SSkMa_d 0.661 0.014 47.284 0.000 0.661 0.828
## SSgerman =~
## SSkDe_a 0.568 0.018 31.610 0.000 0.568 0.643
## SSkDe_b 0.553 0.022 24.924 0.000 0.553 0.523
## SSkDe_c 0.494 0.016 31.469 0.000 0.494 0.640
## SSkDe_d 0.553 0.015 38.039 0.000 0.553 0.758
## SozInt =~
## SBezMs_a 0.544 0.016 34.545 0.000 0.544 0.757
## SBezMs_b 0.495 0.019 25.382 0.000 0.495 0.553
## SBezMs_c 0.524 0.020 26.359 0.000 0.524 0.573
## SBezMs_d 0.429 0.018 24.247 0.000 0.429 0.530
## Abilities =~
## wle_lesen 0.830 0.024 34.203 0.000 0.830 0.687
## wle_hoeren 0.618 0.021 29.300 0.000 0.618 0.597
## wle_mathe 0.909 0.022 40.680 0.000 0.909 0.809
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## SSmath ~~
## SSgerman 0.392 0.022 18.006 0.000 0.392 0.392
## SozInt 0.212 0.024 8.711 0.000 0.212 0.212
## Abilities 0.529 0.019 27.358 0.000 0.529 0.529
## SSgerman ~~
## SozInt 0.263 0.025 10.370 0.000 0.263 0.263
## Abilities 0.427 0.023 18.929 0.000 0.427 0.427
## SozInt ~~
## Abilities 0.167 0.026 6.448 0.000 0.167 0.167
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .SSkMa_a 0.326 0.013 24.737 0.000 0.326 0.378
## .SSkMa_b 0.798 0.024 32.781 0.000 0.798 0.676
## .SSkMa_c 0.313 0.011 28.688 0.000 0.313 0.478
## .SSkMa_d 0.200 0.009 21.359 0.000 0.200 0.314
## .SSkDe_a 0.457 0.016 27.929 0.000 0.457 0.587
## .SSkDe_b 0.815 0.026 31.736 0.000 0.815 0.727
## .SSkDe_c 0.350 0.012 28.040 0.000 0.350 0.590
## .SSkDe_d 0.227 0.011 20.899 0.000 0.227 0.426
## .SBezMs_a 0.221 0.013 17.345 0.000 0.221 0.427
## .SBezMs_b 0.555 0.019 29.542 0.000 0.555 0.694
## .SBezMs_c 0.560 0.019 28.764 0.000 0.560 0.671
## .SBezMs_d 0.472 0.016 30.322 0.000 0.472 0.719
## .wle_lesen 0.770 0.030 25.848 0.000 0.770 0.528
## .wle_hoeren 0.690 0.023 30.039 0.000 0.690 0.643
## .wle_mathe 0.437 0.027 16.471 0.000 0.437 0.346
## SSmath 1.000 1.000 1.000
## SSgerman 1.000 1.000 1.000
## SozInt 1.000 1.000 1.000
## Abilities 1.000 1.000 1.000semPlot::semPaths(object = fit, what = "est")
To account for the non-normal distribution of the questionnaire items and the small sample, the DWLS estimator was used and the \(X^2\) statistic was mean and variance adjusted (e.g. chapter 11 in Hancock and Mueller (2013)):
! limited information approach; FIML, Bayesian SEM is possible in Mplus
datenLV[,c("SSkMa_a",
"SSkMa_b",
"SSkMa_c",
"SSkMa_d",
"SSkDe_a",
"SSkDe_b",
"SSkDe_c",
"SSkDe_d",
"SBezMs_a",
"SBezMs_b",
"SBezMs_c",
"SBezMs_d")] <-
lapply(datenLV[,c("SSkMa_a",
"SSkMa_b",
"SSkMa_c",
"SSkMa_d",
"SSkDe_a",
"SSkDe_b",
"SSkDe_c",
"SSkDe_d",
"SBezMs_a",
"SBezMs_b",
"SBezMs_c",
"SBezMs_d")], ordered)
head(datenLV$SSkMa_a)## [1] 3 3 3 4 3 4
## Levels: 1 < 2 < 3 < 4firstorderCFA <- '
SSmath =~ SSkMa_a + SSkMa_b + SSkMa_c + SSkMa_d
SSgerman =~ SSkDe_a + SSkDe_b + SSkDe_c + SSkDe_d
SozInt =~ SBezMs_a + SBezMs_b + SBezMs_c + SBezMs_d
Abilities =~ wle_lesen + wle_hoeren + wle_mathe
'
# identification: Marker variable method
fit <- sem(firstorderCFA, data = datenLV,
ordered = c("SSkMa_a",
"SSkMa_b",
"SSkMa_c",
"SSkMa_d",
"SSkDe_a",
"SSkDe_b",
"SSkDe_c",
"SSkDe_d",
"SBezMs_a",
"SBezMs_b",
"SBezMs_c",
"SBezMs_d"))
summary(fit, standardized = TRUE, fit.measures=TRUE)## lavaan 0.6-8 ended normally after 37 iterations
##
## Estimator DWLS
## Optimization method NLMINB
## Number of model parameters 63
##
## Used Total
## Number of observations 2566 3005
##
## Model Test User Model:
## Standard Robust
## Test Statistic 1091.263 1391.590
## Degrees of freedom 84 84
## P-value (Chi-square) 0.000 0.000
## Scaling correction factor 0.795
## Shift parameter 19.352
## simple second-order correction
##
## Model Test Baseline Model:
##
## Test statistic 39677.932 22840.150
## Degrees of freedom 105 105
## P-value 0.000 0.000
## Scaling correction factor 1.741
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.975 0.942
## Tucker-Lewis Index (TLI) 0.968 0.928
##
## Robust Comparative Fit Index (CFI) NA
## Robust Tucker-Lewis Index (TLI) NA
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.068 0.078
## 90 Percent confidence interval - lower 0.065 0.074
## 90 Percent confidence interval - upper 0.072 0.082
## P-value RMSEA <= 0.05 0.000 0.000
##
## Robust RMSEA NA
## 90 Percent confidence interval - lower NA
## 90 Percent confidence interval - upper NA
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.057 0.057
##
## Parameter Estimates:
##
## Standard errors Robust.sem
## Information Expected
## Information saturated (h1) model Unstructured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## SSmath =~
## SSkMa_a 1.000 0.837 0.837
## SSkMa_b 0.837 0.018 46.522 0.000 0.701 0.701
## SSkMa_c 0.965 0.015 66.163 0.000 0.807 0.807
## SSkMa_d 1.059 0.015 70.858 0.000 0.886 0.886
## SSgerman =~
## SSkDe_a 1.000 0.682 0.682
## SSkDe_b 0.975 0.031 31.326 0.000 0.665 0.665
## SSkDe_c 1.084 0.029 37.812 0.000 0.740 0.740
## SSkDe_d 1.184 0.029 40.916 0.000 0.808 0.808
## SozInt =~
## SBezMs_a 1.000 0.821 0.821
## SBezMs_b 0.745 0.030 25.241 0.000 0.611 0.611
## SBezMs_c 0.836 0.032 26.338 0.000 0.686 0.686
## SBezMs_d 0.781 0.031 24.827 0.000 0.641 0.641
## Abilities =~
## wle_lesen 1.000 0.811 0.672
## wle_hoeren 0.731 0.035 20.902 0.000 0.593 0.573
## wle_mathe 1.196 0.048 24.758 0.000 0.970 0.863
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## SSmath ~~
## SSgerman 0.273 0.014 19.965 0.000 0.478 0.478
## SozInt 0.166 0.017 9.494 0.000 0.241 0.241
## Abilities 0.367 0.018 19.885 0.000 0.541 0.541
## SSgerman ~~
## SozInt 0.165 0.015 10.840 0.000 0.295 0.295
## Abilities 0.253 0.016 16.022 0.000 0.457 0.457
## SozInt ~~
## Abilities 0.103 0.018 5.751 0.000 0.155 0.155
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .SSkMa_a 0.000 0.000 0.000
## .SSkMa_b 0.000 0.000 0.000
## .SSkMa_c 0.000 0.000 0.000
## .SSkMa_d 0.000 0.000 0.000
## .SSkDe_a 0.000 0.000 0.000
## .SSkDe_b 0.000 0.000 0.000
## .SSkDe_c 0.000 0.000 0.000
## .SSkDe_d 0.000 0.000 0.000
## .SBezMs_a 0.000 0.000 0.000
## .SBezMs_b 0.000 0.000 0.000
## .SBezMs_c 0.000 0.000 0.000
## .SBezMs_d 0.000 0.000 0.000
## .wle_lesen 0.144 0.024 6.032 0.000 0.144 0.119
## .wle_hoeren 0.148 0.021 7.186 0.000 0.148 0.143
## .wle_mathe 0.151 0.022 6.788 0.000 0.151 0.134
## SSmath 0.000 0.000 0.000
## SSgerman 0.000 0.000 0.000
## SozInt 0.000 0.000 0.000
## Abilities 0.000 0.000 0.000
##
## Thresholds:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## SSkMa_a|t1 -1.438 0.037 -39.175 0.000 -1.438 -1.438
## SSkMa_a|t2 -0.828 0.028 -29.478 0.000 -0.828 -0.828
## SSkMa_a|t3 0.074 0.025 3.000 0.003 0.074 0.074
## SSkMa_b|t1 -1.072 0.031 -34.946 0.000 -1.072 -1.072
## SSkMa_b|t2 -0.442 0.026 -17.229 0.000 -0.442 -0.442
## SSkMa_b|t3 0.185 0.025 7.417 0.000 0.185 0.185
## SSkMa_c|t1 -1.758 0.045 -38.947 0.000 -1.758 -1.758
## SSkMa_c|t2 -1.088 0.031 -35.229 0.000 -1.088 -1.088
## SSkMa_c|t3 -0.014 0.025 -0.553 0.581 -0.014 -0.014
## SSkMa_d|t1 -1.801 0.047 -38.664 0.000 -1.801 -1.801
## SSkMa_d|t2 -1.109 0.031 -35.599 0.000 -1.109 -1.109
## SSkMa_d|t3 -0.044 0.025 -1.776 0.076 -0.044 -0.044
## SSkDe_a|t1 -1.575 0.040 -39.504 0.000 -1.575 -1.575
## SSkDe_a|t2 -0.787 0.028 -28.379 0.000 -0.787 -0.787
## SSkDe_a|t3 0.228 0.025 9.111 0.000 0.228 0.228
## SSkDe_b|t1 -1.090 0.031 -35.260 0.000 -1.090 -1.090
## SSkDe_b|t2 -0.420 0.026 -16.448 0.000 -0.420 -0.420
## SSkDe_b|t3 0.308 0.025 12.218 0.000 0.308 0.308
## SSkDe_c|t1 -1.816 0.047 -38.552 0.000 -1.816 -1.816
## SSkDe_c|t2 -1.175 0.032 -36.630 0.000 -1.175 -1.175
## SSkDe_c|t3 0.081 0.025 3.276 0.001 0.081 0.081
## SSkDe_d|t1 -2.003 0.055 -36.643 0.000 -2.003 -2.003
## SSkDe_d|t2 -1.244 0.033 -37.544 0.000 -1.244 -1.244
## SSkDe_d|t3 0.043 0.025 1.737 0.082 0.043 0.043
## SBezMs_a|t1 -2.033 0.056 -36.257 0.000 -2.033 -2.033
## SBezMs_a|t2 -1.294 0.034 -38.099 0.000 -1.294 -1.294
## SBezMs_a|t3 -0.012 0.025 -0.474 0.636 -0.012 -0.012
## SBezMs_b|t1 -1.495 0.038 -39.394 0.000 -1.495 -1.495
## SBezMs_b|t2 -0.808 0.028 -28.930 0.000 -0.808 -0.808
## SBezMs_b|t3 0.243 0.025 9.702 0.000 0.243 0.243
## SBezMs_c|t1 -1.475 0.037 -39.329 0.000 -1.475 -1.475
## SBezMs_c|t2 -0.999 0.030 -33.512 0.000 -0.999 -0.999
## SBezMs_c|t3 -0.278 0.025 -11.078 0.000 -0.278 -0.278
## SBezMs_d|t1 -1.723 0.044 -39.140 0.000 -1.723 -1.723
## SBezMs_d|t2 -1.183 0.032 -36.742 0.000 -1.183 -1.183
## SBezMs_d|t3 -0.456 0.026 -17.735 0.000 -0.456 -0.456
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .SSkMa_a 0.300 0.300 0.300
## .SSkMa_b 0.509 0.509 0.509
## .SSkMa_c 0.348 0.348 0.348
## .SSkMa_d 0.214 0.214 0.214
## .SSkDe_a 0.535 0.535 0.535
## .SSkDe_b 0.558 0.558 0.558
## .SSkDe_c 0.453 0.453 0.453
## .SSkDe_d 0.348 0.348 0.348
## .SBezMs_a 0.326 0.326 0.326
## .SBezMs_b 0.626 0.626 0.626
## .SBezMs_c 0.530 0.530 0.530
## .SBezMs_d 0.588 0.588 0.588
## .wle_lesen 0.801 0.032 25.318 0.000 0.801 0.549
## .wle_hoeren 0.720 0.025 28.933 0.000 0.720 0.672
## .wle_mathe 0.323 0.033 9.688 0.000 0.323 0.255
## SSmath 0.700 0.015 47.431 0.000 1.000 1.000
## SSgerman 0.465 0.019 24.768 0.000 1.000 1.000
## SozInt 0.674 0.028 24.229 0.000 1.000 1.000
## Abilities 0.658 0.039 16.728 0.000 1.000 1.000
##
## Scales y*:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## SSkMa_a 1.000 1.000 1.000
## SSkMa_b 1.000 1.000 1.000
## SSkMa_c 1.000 1.000 1.000
## SSkMa_d 1.000 1.000 1.000
## SSkDe_a 1.000 1.000 1.000
## SSkDe_b 1.000 1.000 1.000
## SSkDe_c 1.000 1.000 1.000
## SSkDe_d 1.000 1.000 1.000
## SBezMs_a 1.000 1.000 1.000
## SBezMs_b 1.000 1.000 1.000
## SBezMs_c 1.000 1.000 1.000
## SBezMs_d 1.000 1.000 1.000semPlot::semPaths(object = fit, what = "est")
3.3.3 SEM (measurement model + structural model)
SEMmodel <- '
# measurement models
SSmath =~ SSkMa_a + SSkMa_b + SSkMa_c + SSkMa_d
SSgerman =~ SSkDe_a + SSkDe_b + SSkDe_c + SSkDe_d
SozInt =~ SBezMs_a + SBezMs_b + SBezMs_c + SBezMs_d
Abilities =~ wle_lesen + wle_hoeren + wle_mathe
# regressions (+2 dummies)
Abilities ~ SSmath + SSgerman + SozInt + Emigr + tr_sex + EHisei
'
# identification: Marker variable method
fit <- sem(SEMmodel, data = datenLV,
ordered = c("SSkMa_a",
"SSkMa_b",
"SSkMa_c",
"SSkMa_d",
"SSkDe_a",
"SSkDe_b",
"SSkDe_c",
"SSkDe_d",
"SBezMs_a",
"SBezMs_b",
"SBezMs_c",
"SBezMs_d"))
summary(fit, standardized = TRUE, fit.measures=TRUE)## lavaan 0.6-8 ended normally after 39 iterations
##
## Estimator DWLS
## Optimization method NLMINB
## Number of model parameters 66
##
## Used Total
## Number of observations 1973 3005
##
## Model Test User Model:
## Standard Robust
## Test Statistic 1505.862 1548.544
## Degrees of freedom 126 126
## P-value (Chi-square) 0.000 0.000
## Scaling correction factor 0.994
## Shift parameter 33.596
## simple second-order correction
##
## Model Test Baseline Model:
##
## Test statistic 27860.196 16284.077
## Degrees of freedom 105 105
## P-value 0.000 0.000
## Scaling correction factor 1.715
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.950 0.912
## Tucker-Lewis Index (TLI) 0.959 0.927
##
## Robust Comparative Fit Index (CFI) NA
## Robust Tucker-Lewis Index (TLI) NA
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.075 0.076
## 90 Percent confidence interval - lower 0.071 0.072
## 90 Percent confidence interval - upper 0.078 0.079
## P-value RMSEA <= 0.05 0.000 0.000
##
## Robust RMSEA NA
## 90 Percent confidence interval - lower NA
## 90 Percent confidence interval - upper NA
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.055 0.055
##
## Parameter Estimates:
##
## Standard errors Robust.sem
## Information Expected
## Information saturated (h1) model Unstructured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## SSmath =~
## SSkMa_a 1.000 0.828 0.828
## SSkMa_b 0.830 0.021 38.856 0.000 0.688 0.688
## SSkMa_c 0.958 0.018 54.715 0.000 0.794 0.794
## SSkMa_d 1.072 0.018 59.008 0.000 0.887 0.887
## SSgerman =~
## SSkDe_a 1.000 0.681 0.681
## SSkDe_b 0.983 0.035 27.891 0.000 0.669 0.669
## SSkDe_c 1.063 0.033 32.359 0.000 0.724 0.724
## SSkDe_d 1.181 0.034 35.089 0.000 0.804 0.804
## SozInt =~
## SBezMs_a 1.000 0.828 0.828
## SBezMs_b 0.768 0.034 22.361 0.000 0.636 0.636
## SBezMs_c 0.808 0.036 22.713 0.000 0.669 0.669
## SBezMs_d 0.766 0.035 21.863 0.000 0.634 0.634
## Abilities =~
## wle_lesen 1.000 0.814 0.693
## wle_hoeren 0.706 0.039 17.878 0.000 0.574 0.572
## wle_mathe 1.130 0.051 21.943 0.000 0.919 0.841
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## Abilities ~
## SSmath 0.383 0.031 12.450 0.000 0.389 0.389
## SSgerman 0.248 0.038 6.571 0.000 0.207 0.207
## SozInt -0.039 0.027 -1.449 0.147 -0.040 -0.040
## Emigr 0.369 0.051 7.206 0.000 0.454 0.178
## tr_sex -0.076 0.039 -1.937 0.053 -0.093 -0.046
## EHisei 0.017 0.001 12.552 0.000 0.021 0.327
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## SSmath ~~
## SSgerman 0.279 0.016 17.807 0.000 0.495 0.495
## SozInt 0.168 0.020 8.468 0.000 0.245 0.245
## SSgerman ~~
## SozInt 0.164 0.018 9.265 0.000 0.290 0.290
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .SSkMa_a 0.000 0.000 0.000
## .SSkMa_b 0.000 0.000 0.000
## .SSkMa_c 0.000 0.000 0.000
## .SSkMa_d 0.000 0.000 0.000
## .SSkDe_a 0.000 0.000 0.000
## .SSkDe_b 0.000 0.000 0.000
## .SSkDe_c 0.000 0.000 0.000
## .SSkDe_d 0.000 0.000 0.000
## .SBezMs_a 0.000 0.000 0.000
## .SBezMs_b 0.000 0.000 0.000
## .SBezMs_c 0.000 0.000 0.000
## .SBezMs_d 0.000 0.000 0.000
## .wle_lesen -1.602 0.152 -10.511 0.000 -1.602 -1.364
## .wle_hoeren -0.980 0.123 -7.956 0.000 -0.980 -0.976
## .wle_mathe -0.892 0.132 -6.777 0.000 -0.892 -0.816
## SSmath 0.000 0.000 0.000
## SSgerman 0.000 0.000 0.000
## SozInt 0.000 0.000 0.000
## .Abilities 0.000 0.000 0.000
##
## Thresholds:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## SSkMa_a|t1 -1.467 0.152 -9.631 0.000 -1.467 -1.467
## SSkMa_a|t2 -0.859 0.151 -5.700 0.000 -0.859 -0.859
## SSkMa_a|t3 0.121 0.150 0.805 0.421 0.121 0.121
## SSkMa_b|t1 -0.723 0.148 -4.881 0.000 -0.723 -0.723
## SSkMa_b|t2 -0.084 0.147 -0.572 0.567 -0.084 -0.084
## SSkMa_b|t3 0.586 0.148 3.961 0.000 0.586 0.586
## SSkMa_c|t1 -1.553 0.154 -10.081 0.000 -1.553 -1.553
## SSkMa_c|t2 -0.882 0.151 -5.852 0.000 -0.882 -0.882
## SSkMa_c|t3 0.225 0.151 1.492 0.136 0.225 0.225
## SSkMa_d|t1 -1.786 0.157 -11.392 0.000 -1.786 -1.786
## SSkMa_d|t2 -1.070 0.153 -7.005 0.000 -1.070 -1.070
## SSkMa_d|t3 0.041 0.152 0.267 0.789 0.041 0.041
## SSkDe_a|t1 -0.456 0.150 -3.050 0.002 -0.456 -0.456
## SSkDe_a|t2 0.292 0.148 1.975 0.048 0.292 0.292
## SSkDe_a|t3 1.367 0.150 9.100 0.000 1.367 1.367
## SSkDe_b|t1 0.211 0.144 1.461 0.144 0.211 0.211
## SSkDe_b|t2 0.899 0.144 6.249 0.000 0.899 0.899
## SSkDe_b|t3 1.672 0.146 11.481 0.000 1.672 1.672
## SSkDe_c|t1 -1.090 0.158 -6.886 0.000 -1.090 -1.090
## SSkDe_c|t2 -0.456 0.152 -3.008 0.003 -0.456 -0.456
## SSkDe_c|t3 0.857 0.153 5.611 0.000 0.857 0.857
## SSkDe_d|t1 -1.218 0.161 -7.542 0.000 -1.218 -1.218
## SSkDe_d|t2 -0.515 0.152 -3.382 0.001 -0.515 -0.515
## SSkDe_d|t3 0.831 0.153 5.418 0.000 0.831 0.831
## SBezMs_a|t1 -1.361 0.160 -8.500 0.000 -1.361 -1.361
## SBezMs_a|t2 -0.596 0.152 -3.917 0.000 -0.596 -0.596
## SBezMs_a|t3 0.722 0.154 4.697 0.000 0.722 0.722
## SBezMs_b|t1 -0.772 0.144 -5.354 0.000 -0.772 -0.772
## SBezMs_b|t2 -0.076 0.143 -0.529 0.597 -0.076 -0.076
## SBezMs_b|t3 1.019 0.146 6.996 0.000 1.019 1.019
## SBezMs_c|t1 -1.196 0.154 -7.759 0.000 -1.196 -1.196
## SBezMs_c|t2 -0.728 0.153 -4.752 0.000 -0.728 -0.728
## SBezMs_c|t3 0.034 0.154 0.223 0.824 0.034 0.034
## SBezMs_d|t1 -1.203 0.164 -7.316 0.000 -1.203 -1.203
## SBezMs_d|t2 -0.650 0.160 -4.056 0.000 -0.650 -0.650
## SBezMs_d|t3 0.091 0.161 0.568 0.570 0.091 0.091
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .SSkMa_a 0.314 0.314 0.314
## .SSkMa_b 0.527 0.527 0.527
## .SSkMa_c 0.370 0.370 0.370
## .SSkMa_d 0.213 0.213 0.213
## .SSkDe_a 0.536 0.536 0.536
## .SSkDe_b 0.552 0.552 0.552
## .SSkDe_c 0.476 0.476 0.476
## .SSkDe_d 0.354 0.354 0.354
## .SBezMs_a 0.315 0.315 0.315
## .SBezMs_b 0.596 0.596 0.596
## .SBezMs_c 0.553 0.553 0.553
## .SBezMs_d 0.598 0.598 0.598
## .wle_lesen 0.718 0.031 22.816 0.000 0.718 0.520
## .wle_hoeren 0.679 0.026 25.904 0.000 0.679 0.673
## .wle_mathe 0.350 0.031 11.178 0.000 0.350 0.293
## SSmath 0.686 0.017 39.302 0.000 1.000 1.000
## SSgerman 0.464 0.022 21.511 0.000 1.000 1.000
## SozInt 0.685 0.032 21.401 0.000 1.000 1.000
## .Abilities 0.378 0.028 13.341 0.000 0.571 0.571
##
## Scales y*:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## SSkMa_a 1.000 1.000 1.000
## SSkMa_b 1.000 1.000 1.000
## SSkMa_c 1.000 1.000 1.000
## SSkMa_d 1.000 1.000 1.000
## SSkDe_a 1.000 1.000 1.000
## SSkDe_b 1.000 1.000 1.000
## SSkDe_c 1.000 1.000 1.000
## SSkDe_d 1.000 1.000 1.000
## SBezMs_a 1.000 1.000 1.000
## SBezMs_b 1.000 1.000 1.000
## SBezMs_c 1.000 1.000 1.000
## SBezMs_d 1.000 1.000 1.000semPlot::semPaths(object = fit, what = "est")
3.4 short: item response theory
wörtliche Anmerkungen, wenn Zeit übrig
3.5 short: longitudinal data / multi-group analysis
wörtliche Anmerkungen, wenn Zeit übrig
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